Indian Statistical Institute, ISI MMATH 2020 PMA Solutions & Discussions

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ISI MMATH 2020 Multiple Choice Questions PMA, Objectives: Solutions and Discussions

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Problem 1

Let $V$ be a finite dimensional vector space and let $W$ be a proper subspace of $V$ . Let $W'$ be another subspace of $V$ such that $V=W\oplus W'$ , i.e. $V=$ span $(W\cup W')$ and $W\cap W'=\{0\}$ . Let $T:V\to V$ be an invertible linear map such that $T(W)\subset W$ . Which of the following statements is necessarily true $?$

(a) $T(W')\subset W'$ ;

(b) $W'\subset T(W')$ ;

(d) $W'\subset \ker(T)$ .

Solution:

The map $T$ is injective, hence $\dim T(W)=\dim W$ and $T$ maps $W$ onto $W$ : $T(W)=W.$ Let $w\in T(W')\cap W.$ There exists $w'\in W'$ such that $Tw'=w.$ But $w\in W=T(W)$ and there exists $v\in W$ such that $Tv=w.$ So,

$Tv=w=Tw'.$

$T$ is injective, hence $v=w'\in W\cap W'=\{0\}.$ So, $w=0.$

Answer (c)

Problem 2

Let $V$ be a finite dimensional real vector space, and let $T:V\to V$ be a linear map such that Range $(T)=\ker(T)$ . Which of the following statements in not necessarily true $?$

(a) $T=0$ ;

(b) $T^2=0$ ;

(d) All eigenvalues of $T$ are equal to $0$ .

Solution:

Consider $T:\mathbb{R}^2\to \mathbb{R}^2$ given by

$T\begin{pmatrix} x \\ y \end{pmatrix}=\begin{pmatrix} 0 \\ x \end{pmatrix}.$

Then $\mbox{Ker}(T)=\{0\}\times \mathbb{R}=\mbox{Range}(T),$ but $T\ne 0.$

Answer (a)

Problem 3

Consider the vector space $\mathbb{R}^n$ equipped with the Euclidean metric $d$ define by

$d(x,y)=\bigg(\sum^n_{i=1}(x_i-y_i)^2\bigg)^1/2$

Let $W$ be a proper subspace of $\mathbb{R}^n$ . Which of the following statements is necessarily true $?$

(a) $W$ is closed.

(b) $W$ is open.

(d) $W$ is neither closed nor open.

Solution:

Let $\{e_1,\ldots,e_n\}$ be a basis in $\mathbb{R}^n$ , such that $\{e_1,\ldots,e_k\}$ is a basis in $W.$ Consider the linear map

$T:\mathbb{R}^n\to \mathbb{R}^{n-k},$

given by $T(\sum^n_{j=1}x_je_j)=(x_{k+1},\ldots,x_n).$ $T$ is continuous. Then $W=\mbox{Ker}(T)$ is closed.

Answer (a)

Problem 4

Let $A$ be a $5\times 5$ real matrix. If $A=(a_{ij})$ , let $A_ij$ denote the cofactor of the entry $a_{ij}$ , for $1\leq i,j\leq 5$ . Let $A$ denote the matrix whose $(i,j)$ -th entry is $A_{ij},1\leq i,j\leq 5$ . Suppose the rank of $A$ is $3$ . What is the rank of $A?$

(a) $1$ ;

(b) $3$ ;

(d) $0$ .

Solution:

Each element of $\hat{A}$ is (up to sign) a $4\times 4$ minor of the matrix $A$ . Since the rank of $A$ is equal to 3, all $4\times 4$ minors of $A$ are equal to $0$ and $\hat{A}$ has rank zero.

Answer (d)

Problem 5

For $n\geq 2$ , the determinant of the $n\times n$ permutation matrix

(a) $(-1)^n$ ;

(b) $(-1)^{n(n-1)/2}$ ;

(d) $1$ .

Solution:

By induction we check that the determinant is equal to $(-1)^{n(n-1)/2}.$ If $n=1,$ then the determinant is equal to $1=(-1)^0.$ Assume the result is true for $n.$ Expanding the determinant of the $(n+1)\times (n+1)$ permutation matrix by the first row we get the answer

$(-1)^n\cdot (-1)^{n(n-1)/2}=(-1)^{n(n+1)/2}$

Answer (b)

Problem 6

Let $M_2(\mathbb{R})$ denote the vector space of all $2\times 2$ matrices over the field of real number i.e.

$M_2(\mathbb{R})=\bigg\{\binom{a\hspace{.6cm}b}{c\hspace{.6cm}d}:a,b,c,d\in\mathbb{R}\bigg\}$

Let $S\subset M_2(\mathbb{R})$ be the subspace defined by

$S=\bigg\{\binom{a\hspace{.6cm}b}{c\hspace{.6cm}d}\in M_2(\mathbb{R}):a+c=0\bigg\}.$

Then the dimension of $S$ is

(a) $1$ ;

(b) $2$ ;

(d) $4$ .

Solution:

The generic element of $S$ is of the form

$\begin{pmatrix} a & b \\ -a & d \end{pmatrix}=a\begin{pmatrix} 1 & 0 \\ -1 & 0 \end{pmatrix}+b\begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}+d\begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix}$

Answer (c)

Problem 7

Let $V$ be a finite dimensional real vector space of dimension $n>1$ and let $W\subset V$ be a subspace of dimension $n-1$ . $A$ linear map from $V$ to $\mathbb{R}$ is called a linear functional on $V$ . Which of the following statements is necessarily true $?$

(a) There does not exist any linear functional on $V$ such that $W$ is the kernel of that linear functional;

(b) $W$ is the kernel of unique linear functional on $V$ ;

(d) There exist a non-zero linear functional on $V$ whose kernel strictly contains $W$ .

Solution:

Let $a\in V\setminus W.$ Then every element $v\in V$ has a unique expansion

$v=w+ta,\ t\in\mathbb{R}.$

Define $f(v)=ct,$ $c\ne 0.$ Then $f\ne 0$ and $\mbox{Ker}(f)=W.$

Answer (c)

Problem 8

Let $<\dots >_n$ denote the standard inner product on the vector space $\mathbb{R}^n$ ,i.e.

$<x,y>_n=\sum^n_{i=1}x,y_i$

for vectors $x,y\in\mathbb{R}^n$ . Let $T:\mathbb{R}^n\to\mathbb{R}^m$ be a linear map such that

$<Tx,Ty>_m=<x,y>_n$

for all $x,y\in\mathbb{R}^n$ . which of the following statements is necessarily true $?$
(a) $n\geq m$ .

(b) $n\leq m$ .

(d) The map $T$ is onto

Solution:

The map $T$ is injective, so it maps the basis of $\mathbb{R}^n$ into a linearly independent subset of $\mathbb{R}^m.$

Answer (b)

Problem 9

Let $V$ be a finite dimensional real inner product space and let $T:V\to V$ be a linear map such that $<TxTy>=<x,y>$ for all $x,y\to V$ . Suppose $W\subset V$ is a proper subspace of $V$ such that $T(W)\subset W$ Define a subspace $W^{\perp}$ of $V$ by

$W^{\perp} := \{v\in V|<v,w>=0 \mbox{ for all } w\in W\}$

(a) $T(W^{\perp})$ is not contained in $W^{\perp}$ .

(b) $T(W^{\perp})$ is contained in $W^{\perp}$ .

(d) $W^{\perp}$ is contained in $T(W^{\perp})$

Solution:

Let $v\in W^\perp.$ There exists $x\in V$ such that $Tx=v.$ For every $w\in W$ we have $Tw\in W$ and

$\langle x,w\rangle =\langle Tx,Tw\rangle=\langle v,Tw\rangle=0.$

So, $x\in W^\perp$ and $v\in T(W^\perp).$

Answer (d)

Problem 10

Let $R$ be a ring with unit such that $a^2=a$ for all $a\in\mathbb{R}$ . Which of the following statements is not neccessarily true $?$
(a) $ab=-ba$ for all $a,b,\in\mathbb{R}$ ,

(b) $a=-a$ for all $a\in\mathbb{R}$ ,

(d) $R=\{0,1\}$ .

Solution:

Let $R$ be the ring of all subsets of a set $S,$ with operations

$A+B=A\Delta B, AB=A\cap B.$

$R$ is an idempotent ring, $R\ne \{0,1\}.$

Answer (d)

Problem 11

Let $S$ be a nonempty set, and let $p(S)$ be the power set of S, i.e. $p(S)=\{A|A\subset S|\}$ . Define a binary operation $\angle$ on $P(S)$ by $A\angle B:=(A\cup B)\setminus (A\cap B)$ for $A,B\in P(S)$ . Which of the following statements is necessarily true $?$

(a) $(P(S),\triangle)$ is not a group as $\triangle$ is not associative.

(b) $(P(S),\triangle)$ is not a group as there is no identity.

(d) $(P(S),\triangle)$ is a non-abelian group.

Solution:

With the operation $\Delta$ , the power set $P(S)$ is an abelian group:

$A\Delta \emptyset =A, A\Delta A=\emptyset,$

$A\Delta B=B\Delta A, (A\Delta B)\Delta C=A\Delta (B\Delta C).$

Answer (c)

Problem 12

Let $I_1,I_2$ be ideals of a commutative ring $\mathbb{R}$ . Define the set

$I_1+I_2:=\{a+b|a\in I_1, b\in I_2\}.$

Which of the following statements in not neccessarily true $?$
(a) $I_1+I_2$ is an ideal of $\mathbb{R}$ .

(b) $I_1\subset I_1+I_2$ .

(d) $|I_1+I_2|=|I_1|\cdot |I_2|$ if $I_1\cap I_2=\{0\}$ and $\mathbb{R}$ is finite.

Solution:

If $I_1=I_2$ with $|I_1|>1,$ then $I_1+I_2=I_1$ and (c) fails.

Answer (c)

Problem 13

Let $I$ be an ideal of a commutative ring $\mathbb{R}$ . Define the set

$\sqrt{I}:=\{a\in\mathbb{R}|\mbox{ There exists }n\geq 1 \mbox{ such that } a^n\in I \}.$

Which of the following statements is neccessarily true $?$
(a) $\sqrt{I}$ is an ideal.

(b) $\sqrt{I}$ is not an ideal.

(d) $\sqrt{I}\subset I$ .

Solution:

We check that $\sqrt{I}$ is an ideal. Let $a,b\in \sqrt{I}.$ Then $a^n,b^m\in I$ for some $n,m\geq 1.$ We have

$(a+b)^{n+m}=\sum^{n+m}_{j=0}{n+m\choose j}a^jb^{n+m-j}.$

If $j\geq n,$ then $a^jb^{n+m-j}=a^n (a^{j-n}b^{n+m-j})\in I.$

If $j<n,$ then $a^jb^{n+m-j}=b^m (a^{j}b^{n-j})\in I.$ In any case $a^jb^{n+m-j}\in I,$ so $(a+b)^{n+m}\in I$ and $a+b\in\sqrt{I}.$

If $a\in \sqrt{I},$ $r\in R,$ then $a^n\in I$ for some $n\geq 1$ and $(ra)^n=r^na^n\in I.$ So, $ra\in\sqrt{I}.$

Answer (a)

Problem 14

How many non-isomorphic group are there of order $15?$

(a) $1$ ;

(b) $2$ ;

(d) $5$ .

Solution:

There is an element $a\in G$ of order 3 and an element $b\in G$ of order 5. They generate cyclic Sylow subgroups $P$ and $Q$ of $G$ . Let $n_3$ be the number of $3-$ Sylow subgroups of $G.$ Then

$n_3| 5, \ n_3\equiv 1 \mod 3.$

It follows that $n_3=1$ and $P$ is a normal subgroup of $G.$ Similarly, $Q$ is a normal subgroup of $G$ . Elements of $P$ commute with elements of $Q,$ and the order of $ab$ is $3\cdot 5=15.$ $G$ is cyclic.

Answer (a)

Problem 15

Suppose $G$ is a group and $a,b\in G$ . which of the following statements is neccessarily true $?$

(a) There exist a positive integer $n$ such that $a^n=b^n$ ;

(b) $(ab)^{-1}=a^{-1}b^{-1}$ ;

(d) None of the above statements is necessarily true.

Solution:

If $(ab)^n=e,$ then

$(ba)^n=b(ab)^nb^{-1}=bb^{-1}=e$

This proves $o(ab)\geq o(ba).$ By symmetry, $o(ab)=o(ba).$

Answer (c)

Problem 16

Let $H_1,H_2$ be distinct subgroups of a finite abelian group $G$ . Define the subgroup $H_1H_2$ by $H_1H_2={h_1h_2|h_1\in H_1,h_2\in H_2}$ .
Which of the following statements is neccessarily true $?$
(a) $|G|\leq |H_1|+|H_2|$ ;

(b) $|G\setminus (H_1\cap H_2)|=|G\subminus H_1|\cdot |G\setminus H_2|$ ,

(d) $|(H_1 H_2)\setminus H_1|=|H_2\setminus (H_1\cup H_2)|$ .

Solution:

Consider the mapping $f:H_1H_2\to H_2/(H_1\cap H_2)$ given by

$f(h_1h_2)=h_2(H_1\cap H_2).$

$f$ is well-defined. Indeed, if

$h_1h_2=h'_1h'_2,$

then $h^{-1}_1h'_1=h^{-1}_2h'_2\in H_1\cap H_2$ and

$h_2(H_1\cap H_2)=h'_2(H_1\cap H_2).$

$f$ is a homomorphism. Its kernel coincides with $H_1:$

$h_2\in H_1\Leftrightarrow h_1h_2\in H_1.$

So, the quotient of $f$ is an isomorphism.

Answer (d)

Problem 17

Let $P>3$ be a prime number and let $s_p$ denote the symmetric group on $P$ symbols. How many $p$ -Sylow subgroup are there in $S_p?$

(a) $1$ ;

(b) $p$ ;

(d) $(P-2)!$ .

Solution:

Every $p-$ Sylow subgroup of $S_p$ is a cyclic group. It contains $(p-1)$ elements of order $p.$ Every element of order $p$ in $S_p$ is given by a permutation $(1,n_2,\ldots,n_p)$ of $\{1,2,\ldots,p\}.$ There are $(p-1)!$ such permutations, so the number of $p-$ Sylow subgroups is $\frac{(p-1)!}{p-1}=(p-2)!.$

Answer (d)

Problem 18

Let $\mathbb{R}$ be a commutative ring with unit and let

$N=\{a\in \mathbb{R}|a^n=0 \mbox{ for some integer } n\geq 0\}.$

Which of the following statements is necessarily true $?$
(a) Any prime of $\mathbb{R}$ contains $N$ ,

(b) $N$ is not an ideal,

(d) $N=\{0\}$ .

Solution:

Let $P$ be a prime ideal in $R$ . For any $a\in N$ we have for some $n\geq 1$

$a^n=aa\ldots a=0\in P.$

The ideal is prime, hence one of the factors is in $P$ , i.e. $a\in P$ and $N \subset P.$

Answer (a)

Problem 19

Let $f:\mathbb{R}\to\mathbb{R}$ be twice continuously differentiable, and suppose $\lim_{x\to\infty}f''(x)=1.$ Which of the following statements is necessarily true $?$

(a) $\lim_{x\to\infty}\frac{f(x)}{x^2}=1$ ,

(b) $\lim_{x\to\infty}\frac{f(x)}{x^2}$ does not exist,

(d) $\lim_{x\to\infty}\frac{f(x)}{x^2}=1\setminus 2$ .

Solution:

Apply Taylor’s expansion:

$f(x)=f(0)+xf'(0)+\int^x_0 (x-t)f''(t)dt.$

$\lim_{x\to\infty}\frac{f(x)}{x^2}=\lim_{x\to\infty}\frac{\int^x_0(x-t)f''(t)dt}{x^2}=$

applying L’Hopital’s rule twice

$=\lim_{x\to\infty}\frac{\int^x_0f''(t)dt}{2x}=\lim_{x\to\infty}\frac{f''(x)}{2}=\frac{1}{2}$

Answer (d)

Problem 20

Let $f_n:[0,1]\to\mathbb{R}$ be define by $f_n(x)=(\cos(\pi x))^{2n}$ . Which of the following statements is true $?$

(a) The sequence $\{f_n\}$ converges uniformly on ${0,1}$ ,

(b) The sequence $\{f_n\}$ converges pointwise on $[0,1]$ to a function $f$ such that $f$ has exactly one point of discontinuity,

(c) The sequence $\{f_n\}$ converges pointwise on $[0,1]$ to a function $f$ such that $f$ has exactly two points of discontinuity,

(d) The sequence $\{f_n\}$ does not converge pointwise on $[0,1]$ .

Solution:

$f_n(0)=f_n(1)=1.$ If $0<x<1,$ then $|\cos(\pi x)|<1$ and $f_n(x)\to 0.$

Answer (c)

Problem 21

Let $f: [-1,1] \to\mathbb{R}$ be a continuous function such that

$\int^1_{-1} f(x) x^{2n}dx=0$

for all $n\geq 0$ .Which of the following statements is necessarily false $?$
(a) $f^1_{-1}f(x)^2 dx=f^1_{-1}f(-x)^2 dx$ .

(b) $\bigg(\sup_{x\in|-1,1|}f(x)\bigg)+\bigg(\inf_{x\in|-1,1|}f(x)\bigg)=0$ .

(d) $f(1\setminus 2)f(-1\setminus 2)\leq 0$ .

Solution:

Consider function $g(x)=\frac{f(x)+f(-x)}{2}.$ $g(x)$ is continuous on $[-1,1]$ and for all $n\geq 0$

$\int^1_{-1}g(x)x^ndx=0.$

Since polynomials are dense in $L^2([-1,1])$ we deduce that $g(x)=0.$ In particular,

$g(0)=f(0)=0.$

Answer (c)

Problem 22

Let $f:\mathbb{R}\to\mathbb{R}$ be a continuous function such that $f(x+1)=f(x)+1$ for all $x\in\mathbb{R}$ . Which of the following statements is necessarily false $?$

(a) $\lim_{x+\infty}\frac{f(x)}{x^{1+e}}=0$ for all $\in>0$ ,

(b) $\lim_{x+\infty}\frac{f(x)}{x}$ does not exist,

(d) $\lim_{x+\infty}\frac{f(x)}{x^{1+e}}=+\infty$ for all $\in >0$ .

Solution:

Let $C=\sup_{x\in[0,1]}|f(x)|.$ Let $n\leq x<n+1.$ Then
$f(x)=f(n+(x-n))=n+f(x-n)$ and

$\frac{f(x)}{x}=\frac{n+f(x-n)}{n+(x-n)}=\frac{1+\frac{f(x-n)}{n}}{1+\frac{x-n}{n}}\to 1, \ n\to\infty$

Answer (b)

Problem 23

Let $\{0,1\}^{\mathbb{N}}$ denote the set of all sequence $\{x_n\}$ such that $x_n\in\{0,1\}$ for all $n\geq$ . Define a map $f:\{0,1\}^{\mathbb{N}}\to\mathbb{R}$ by

$f(\{x_n\}):=\sum^{\infty}_{n=1}\frac{x_n}{2^n}.$

Which of the following statements is true $?$
(a) The map $f$ is one-to-one and onto from $\{0,1\}^{\mathbb{N}}$ to $[0,1]$ .

(b) The map $f$ is one-to-one and onto from $\{0,1\}^{\mathbb{N}}$ to $[0,1)$ .

(d) The map $f$ is onto from $\{0,1\}^{\mathbb{N}}$ to $[0,1]$ and $|f^{-1}(1)|=2$ .

Solution:

Every real number $y\in [0,1]$ has a binary expansion

$y=\sum^\infty_{n=1}\frac{x_n}{2^n}, \ x_n\in \{0,1\}, n\geq 1.$

The map $f:\{0,1\}^{\mathbb{N}}\to [0,1]$ is onto. We observe that

$\frac{1}{2}=\sum^\infty_{n=2}\frac{1}{2^n}$

Equivalently,

$f(1,0,0,0,\ldots)=f(0,1,1,1,1,\ldots)=\frac{1}{2}.$

Answer (c)

Problem 24

Let $f:[0,\infty)\to\mathbb{R}$ be a monotone increasing functions, and define $f_n:[0,\infty)\to\mathbb{R}$ by

$f_n(x)=f(x+n),x\in[0,\infty)$

for all $n\geq 1$ . Suppose that for some $x_0\in[0,\infty)$ , the limit $\lim_{n\to\infty}f_n(x_0)$ , exists. Which of the following statements is necessarily false $?$
(a) The sequence $\{f_n\}$ converges pointwise on $[0,\infty)$ .

(b) The sequence $\{f_n\}$ converges uniformly on $[0,\infty)$ .

(d) The function $f$ is unbounded on $[0,\infty)$ .

Solution:

Let $y=\lim_{n\to\infty}f_n(x_0)=\lim_{n\to\infty}f(x_0+n)=\sup_{n\geq 1}f(x_0+n).$ Let $x\geq 0.$ Find $n\geq 1$ such that $x_0+n\geq x.$ Then

$f(0)\leq f(x)\leq f(x_0+n)\leq y.$

$f$ is bounded.

Answer (d)

Problem 25

Let $X,Y$ be set and let $f:X\to Y$ be a function. Let $\{S_i\}_{i\in I}$ be a family of subsets of $X$ ,i.e. $S_i\subset X$ for all $i\in I$ , Where $I$ is an index set.
Which of the following statements is not necessarily true $?$

(a) $F(\cup_i\in IS_i)\subset \cup_{i\in I}f(S_i)$ .

(b) $F(\cup_i\in IS_i)\supset\cap_{i\in I}f(S_i)$ .

(d) $F(\cup_i\in IS_i)\supset \cup_{i\in I}f(S_i)$ .

Solution:

Assume $f(x)=y_0$ for all $x\in X.$ Let $S_1,S_2\subset X$ be non-empty sets such that $S_1\cap S_2=\emptyset.$ Then

$f(S_1)\cap f(S_2)=\{y_0\},$

while

$f(S_1\cap S_2)=f(\emptyset)=\emptyset.$

Answer (d)

Problem 26

Let $S$ be the set of all those nonnegative real numbers $\alpha$ with the following property: if $\{x_n\}$ is a sequence of nonnegative real number such that $\sum^{infty}_{n=1}x_n<+\infty$ , then we also have $\sum^{\infty}_{n=1}\frac{\sqrt{x_n}}{n^{\alpha}}<+\infty$ . Which of the following statements is true $?$

( a) $S=\emptyset$ .

(b) $S\supset(1/ 4),\infty$ .

(d) $S\subset (3/ 4,\infty)$ .

Solution:

By Cauchy-Schwarz inequality

$\sum^\infty_{n=1}\frac{\sqrt{x_n}}{n^\alpha}\leq \sqrt{\sum^\infty_{n=1}x_n}\sqrt{\sum^\infty_{n=1}\frac{1}{n^{2\alpha}}}$

The right-hand side is finite when $2\alpha>1,$ i.e. $\alpha>\frac{1}{2}.$

Answer (c)

Problem 27

Let $X$ be a finite set. Let $P(X)$ be the power set of $X$ , i.e. the set whose elements are all subset of $X$ . Which of the following defines a metric on the power set $P(X)?$

(a) $d(V,W)=|(V\cup W)\setminus(V\cap W)|$ .

(b) $d(V,W)=|V\cap W|$ .

(d) $d(V,W)=|V\cup W|$ .

Solution:

The metric is given by $d(V,W)=|(V\cup W)\setminus (V\cap W)|=|V\Delta W|.$ Indeed, from inclusion

$V\Delta W\subset (V\Delta Z)\cup (Z\Delta W)$

it follows that

$d(V,W)=|V\Delta W|\leq |V\Delta Z|+|Z\Delta W|=d(V,Z)+d(Z,W).$

Further,

$d(V,W)=|V\Delta W|=|W\Delta V|=d(W,V)$

and

$d(V,W)=0\Leftrightarrow V\Delta W=\emptyset \Leftrightarrow V=W.$

Answer (a)

Problem 28

The tangent line to the curve $2_x^6+y^4=9_{xy}$ at the point $(1,2)$ has slope

(a) $3/23$ ,

(b) $6/23$ ,

(d) $4/7$ .

Solution:

The slope of the tangent line is $\frac{dy}{dx}.$ Differentiating the equation of the curve we get

$12x+4y^3\frac{dy}{dx}=9y+9x\frac{dy}{dx}$

At (1,2) we obtain

$\frac{dy}{dx}=\frac{9y-12x}{4y^3-9x}=\frac{6}{23}$

Answer (b)

Problem 29

Consider the following statements:

1. If $\sum_n a_n$ and $\sum_n b_n$ are convergent, then $\sum_n a_n b_n$ is convergent.
2. If $\sum_n a_n$ is convergent and $\sum_n b_n$ is absolutely convergent, then $\sum_n a_nb_n$ is absolutely convergent.
3. If $a_n\geq 0$ for all $n$ , $\sum_n a_n$ is convergent, and $\{b_n\}$ is bounded sequence, then $\sum_n a_nb_n$ is absolutely convergent, then $\sum_n a_nb_n$ is absolutely convergent.
Which of the following statements is true $?$

(a) All of the statements are true,

(b) Statement is true but statement is false,

(d) Only statements and are true.

Solution:

Statement (i) is false: consider the series $\sum_n \frac{(-1)^n}{\sqrt{n}}.$ The series is convergent by the alternating series test. Take $a_n=b_n=\frac{(-1)^n}{\sqrt{n}}.$ Then both series $\sum_na_n$ and $\sum_n b_n$ are convergent, but the series $\sum_n a_nb_n=\sum_n \frac{1}{n}$ is divergent.

Statement (ii) is true. The sequence $a_n$ converges to zero, so it is bounded. Hence,

Statement (iii) is true by the same reasons: $\sum_n a_n$ is absolutely convergent, sequence $(b_n)$ is bounded.

Answer (d)

Problem 30

Consider the metric space $(\mathbb{N}=\mathbb{N}\cup\{\infty\},d)$ , Where the metric $d$ is define by $d(m,n)=|\frac{1}{m}-\frac{1}{n}|$ for $m,n\in \mathbb{N}$ , and $d(n,\infty)=1/n$ for $n\in\mathbb{N}$ . Let $f:\mathbb{N}\to\mathbb{R}$ be a continuous function between metric space (Where $\mathbb{R}$ is equipped with its usual metric). Which of the following statements is necessarily false $?$
(a) The metric space $\mathbb{N}$ is compact,

(b) The function $f$ is unbounded,

(d) For any $x\in\mathbb{R}$ , the set $f^{-1}(\{x\})$ is compact.

Solution:

We prove that the metric space $\overline{\mathbb{N}}$ is compact. Indeed, let $\{V_\alpha\}_\alpha$ be an open cover of $\overline{\mathbb{N}}.$ Let $\infty\in V_{\alpha_0}.$ There exists $\epsilon>0$ such that $B(\infty,\epsilon)\subset V_{\alpha_0}.$ If $n>\frac{1}{\epsilon},$ then $d(n,\infty)=\frac{1}{n}<\epsilon$ and $n\in V_{\alpha_0}.$ So, $V_{\alpha_0}$ contains all points of $\overline{\mathbb{N}}$ except possibly a finite number of points. The remaining set of points can be covered by finite number of sets $V_\alpha.$ The space $\overline{\mathbb{N}}$ is compact. Continuous function $f:\overline{\mathbb{N}}\to \mathbb{R}$ defined on a compact space is bounded.

Answer (b)