# Real Analysis: Short Questions and MCQs

## Short questions

- What is the difference between rational and irrational numbers?
- Is there a rational number exists between any two rational numbers.
- Is there a real number exists between any two real numbers.
- Is the set of rational numbers countable?
- Is the set of real numbers countable?
- Give an example of sequence, which is bounded but not convergent.
- Is every bounded sequence convergent?
- Is product of two convergent sequences convergent?
- Give an examples of two divergence sequences, whose sum is convergent.
- Prove that $\left\{\frac{1}{n+1} \right\}$ is decreasing sequence.
- Is the sequence $\left\{\frac{n+2}{n+1} \right\}$ is increasing or decreasing?
- If the sequence $\{x_n\}$ converges to 5 and $\{y_n\}$ converges to 2. Then find $\lim_{n\to\infty z_n}$, where $z_n=x_n-2y_n$.
- If the sequence $\{x_n\}$ converges to 3 and $\{y_n\}$ converges to 4. Then find $\lim_{n\to\infty z_n}$, where $x_n=2y_n-3z_n$.
- Give an example to prove that bounded sequence may not convergent.
- Prove that every convergent sequence is bounded.

## Multiple choice questions (MCQs)

### Real Number System

1. What is not true about number zero.

- (A) Even
- (B) Positive
- (C) Additive identity
- (D) Additive inverse of zero

(B): zero is neither positive not negative

2. Which one of them is not interval.

- (A) $(1,2)$
- (B) $\left(\frac{1}{2},\frac{1}{3} \right)$
- (C) $[3. \pi]$
- (D) $(2\pi,180)$

(B): In interval $(a,b)$, $a<b$ but $\frac{1}{2}>\frac{1}{3}$.

3. A number which is neither even nor odd is

- (A) 0
- (B) 2
- (C) $2n$ such that $n \in \mathbb{Z}$
- (D) $2\pi$

(D): Integers can only be even or odd but $2\pi$ is not an integer.

4. A number which is neither positive nor negative is

- (A) 0
- (B) 1
- (C) $\pi$
- (D) None of these

(A): zero is number which is neither positive nor negative .

5. Concept of the divisibility only exists in set of …………..

- (A) natural numbers
- (B) integers
- (C) rational numbers
- (D) real numbers

(B): In integers, we define divisibility rugosely

6. If a real number is not rational then it is ……………

- (A) integer
- (B) algebraic number
- (C) irrational number
- (D) complex numbers

(C): Real numbers can be partitioned into rational and irrational.

7. Which of the following numbers is not irrational.

- (A) $\pi$
- (B) $\sqrt{2}$
- (C) $\sqrt{3}$
- (D) 7

(D): Its easy to see

8. A set $A$ is said to be countable if there exists a function $f:A\to \mathbb{N}$ such that

- (A) $f$ is bijective
- (B) $f$ is surjective
- (C) $f$ is identity map
- (D) None of these

(A): By definition of countable set, it must be bijective.

9. Let $A=\{x| x\in \mathbb{N} \wedge x^2 \leq 7 \} \subset \mathbb{N}$. Then supremum of $A$ is

- (A) 7
- (B) 3
- (C) 2
- (D) does not exist

(C): In tabular form $A=\{1, 2 \}$ and set of upper bouds is $\{2,3,4,... \}$. Now supremum is least upper bound $2$.

### Sequence of Numbers

1. A convergent sequence has only ……………. limit(s).

- (A) one
- (B) two
- (C) three
- (D) None of these

(A): limit of the sequence, if it exist, is unique.

2. A sequence $\{s_n\}$ is said to be bounded if

- (A) there exists number $\lambda$ such that $|s_n|<\lambda$ for all $n\in\mathbb{Z}$.
- (B) there exists real number $p$ such that $|s_n|<p$ for all $n\in\mathbb{Z}$.
- (C) there exists positive real number $s$ such that $|s_n|<s$ for all $n\in\mathbb{Z^+}$.
- (D) the term of the sequence lies in a vertical strip of finite width.

(C): It is a definition of bounded sequence.

3. If the sequence is convergent then

- (A) it has two limits.
- (B) it is bounded.
- (C) it is bounded above but may not be bounded below.
- (D) it is bounded below but may not be bounded above.

(B): If a sequence of real numbers is convergent, then it is bounded.

4. A sequence $\{(-1)^n\}$ is

- (A) convergent.
- (B) unbounded.
- (C) divergent.
- (D) bounded.

(D): As $|(-1)^n| = 1 < 1.1$ for all $n \in \mathbb{N}, therefore it is bounded.$

5. A sequence $\left\{\dfrac{1}{n} \right\}$ is

- (A) bounded.
- (B) unbounded.
- (C) divergent.
- (D) None of these.

(A): As $\left\{\dfrac{1}{n} \right\}$ is convergent, it is bounded or it is easy to see $\left|\dfrac{1}{n} \right| \leq 1$ for all $n \in \mathbb{N}$.

6. A sequence $\{s_n\}$ is said be Cauchy if for $\epsilon>0$, there exists positive integer $n_0$ such that

- (A) $|s_n-s_m|<\epsilon$ for all $n,m>0$.
- (B) $|s_n-s_m|<n_0$ for all $n,m>\epsilon$.
- (C) $|s_n-s_m|<\epsilon$ for all $n,m>n_0$.
- (D) $|s_n-s_m|<\epsilon$ for all $n,m<n_0$.

(C): Definition of Cauchy sequence.

7. Every Cauchy sequence has a ……………

- (A) convergent subsequence.
- (B) increasing subsequence.
- (C) decreasing subsequence.
- (D) positive subsequence.

(A): Every Cauchy sequence has a convergent subsequence.

8. A sequence of real number is Cauchy iff

- (A) it is bounded
- (B) it is convergent
- (C) it is positive term sequence
- (D) it is convergent but not bounded.

(B): Cauchy criterion for convergence of sequences.

9. Let $\{s_n\}$ be a convergent sequence. If $\lim_{n\to\infty}s_n=s$, then

- (A) $\lim_{n\to\infty}s_{n+1}=s+1$
- (B) $\lim_{n\to\infty}s_{n+1}=s$
- (C) $\lim_{n\to\infty}s_{n+1}=s+s_1$
- (D) $\lim_{n\to\infty}s_{n+1}=s^2$.

(B): If $n\to\infty$, then $n+1\to\infty$ too.

10. Every convergent sequence has …………….. one limit.

- (A) at least
- (B) at most
- (C) exactly
- (D) none of these

(C): Every convergent sequence has unique limit.

11. If the sequence is decreasing, then it …………….

- (A) converges to its infimum.
- (B) diverges.
- (C) may converges to its infimum
- (D) is bounded.

(C): If the sequence is bounded and decreasing, then it convergent.

12. If the sequence is increasing, then it …………….

- (A) converges to its supremum.
- (B) diverges.
- (C) may converges to its supremum.
- (D) is bounded.

(C): If the sequence is bounded and decreasing, then it convergent.

13. If a sequence converges to $s$, then ………….. of its sub-sequences converges to $s$.

- (A) each
- (B) one
- (C) few
- (D) none

(A): Every subsequence of convergent sequence converges to the same limit.

14. If two sub-sequences of a sequence converge to two different limits, then a sequence ……………

- (A) may convergent.
- (B) may divergent.
- (C) is convergent.
- (D) is divergent.

(D): Every subsequence of convergent sequence converges to the same limit.

### Series of Numbers

1. A series $\sum_{n=1}^\infty a_n$ is said to be convergent if the sequence $\{ s_n \}$, where ………………

- (A) $s_n=\sum_{n=1}^\infty a_n$ is convergent.
- (B) $s_n=\sum_{k=1}^n a_k$ is convergent.
- (C) $s_n=\sum_{k=1}^n a_n$ is convergent.
- (D) $s_n=\sum_{k=1}^n a_k$ is divergent.

(B): Series is convergent if its sequence of partial sume is convergent.

2. If $\sum_{n=1}^\infty a_n$ converges then ………………………

- (A) $\lim_{n\to \infty} a_n=0$.
- (B) $\lim_{n\to \infty} a_n=1$.
- (C) $\lim_{n\to \infty} a_n \neq 0$
- (D) $\lim_{n\to \infty} a_n$ exists.

(A)

3. If $\lim_{n\to \infty} a_n \neq 0$, then $\sum_{n=1}^\infty a_n$ ………………………

- (A) is convergent.
- (B) may convergent.
- (C) is divergent
- (D) is bounded.

(C): It is called divergent test

4. A series $\sum_{n=1}^\infty \left( 1+\frac{1}{n} \right)$ is ………………..

- (A) convergent.
- (B) divergent.
- (C) constant.
- (D) none of these

(B): As $\lim_{n\to \infty}\,\left( 1+\frac{1}{n} \right)=1\ne 0$, therefore by divergent test, the given series is divergent.

5. Let $\sum a_n$ be a series of non-negative terms. Then it is convergent if its sequence of partial sum ……………

- (A) is bounded.
- (B) may bounded.
- (C) is unbounded.
- (D) is divergent.

(A): If $\sum a_n$ is a non-negative terms series, then its sequence of partial sum is increasing. A monotone sequence of partial sume is convergent, if it is bounded.

6. If $\lim_{n\to\infty} a_n=0$, then $\sum a_n$ …………….

- (A) is convergent.
- (B) is divergent.
- (C) may or may not convergent
- (D) none of these

(C): If $\sum a_n$ is convergent, then $\lim_{n\to\infty} a_n=0$ but converse may not true. e.g., $\sum \frac{1}{n}$ is divergent.

7. A series $\sum \frac{1}{n^p}$ is convergent if

- (A) $p\leq 1$.
- (B) $p\geq 1$.
- (C) $p<1$.
- (D) $p>1$.

(D): The p-series test, it can be proved easily by Cauchy condensation test.

8- If a sequence $\{a_n\}$ is convergent then the series $\sum a_n$ …………….

- (A) is convergent.
- (B) is divergent.
- (C) may or may not convergent
- (D) none of these

(C): The p-series test, it can be proved easily by Cauchy condensation test.

9. An alternating series $\sum (-1)^n a_n$, where $a_n\geq 0$ for all $n$, is convergent if

- (A) $\{a_n\}$ is convergent.
- (B) $\{a_n\}$ is decreasing.
- (C) $\{a_n\}$ is bounded.
- (D) $\{a_n\}$ is decreasing and $\lim a_n=0$.

(B): Its called alternating series test.

10. An series $\sum a_n$ is said to be absolutely convergent if

- (A) $\left| \sum a_n \right|$ is convergent.
- (B) $\left| \sum a_n \right|$ is convergent but $\sum a_n$ is divergent.
- (C) $\sum |a_n|$ is convergent.
- (D) $\sum |a_n|$ is divergent but $\sum a_n$ is convergent.

(C): It is definition of absolutely convergent.

11. A series $\sum a_n$ is convergent if and only if ………………… is convergent

- (A) $\{\sum_{k=1}^{\infty}a_k \}$
- (B) $\{\sum_{k=1}^{n}a_k \}$
- (C) $\{\sum_{n=1}^{\infty}a_k \}$
- (D) $\{ a_n \}$

(B): By definition, a series is convergent if its sequence of partial sum is convergent.

### Limit of functions

1. A number $L$ is called limit of the function $f$ when $x$ approaches to $c$ if for all $\varepsilon>0$, there exist $\delta>0$ such that ……… whenever $0<|x-c|<\delta$.

- (A) $|f(x)-L| > \varepsilon$
- (B) $|f(x)-L| < \varepsilon$
- (C) $|f(x)-L| \leq \varepsilon$
- (D) $|f(x)-L| \geq \varepsilon$

(B): It is a definition of limit of functions.

2. If $\lim_{x \to c}f(x)=L$, then ………….. sequence $\{x_n\}$ such that $x_n \to c$, when $n\to \infty$, one has $\lim_{n \to \infty}f(x_n)=L$.

- (A) for some
- (B) for every
- (C) for few
- (D) none of these

(B)

3. Let $f(x)=\frac{x^2-5x+6}{x-3}$, then $\lim_{x\to 3}f(x)=$………..

- (A) $-1$
- (B) $0$
- (C) $1$
- (D) doesn't exist.

(C): $\lim_{x\to 3}f(x)=\frac{x^2-5x+6}{x-3}=\lim_{x\to 3}\frac{(x-2)(x-3)}{x-3}$ $=\lim_{x\to 3}(x-2) = 1$.

### Riemann Integrals

1. Which one is not partition of interval $[1,5]$.

- (A) $\{1,2,3,5 \}$
- (B) $\{1,3,3.5,5 \}$
- (C)$\{1,1.1,5 \}$
- (D) $\{1,2.1,3,4,5.5 \}$

(D): All points must be between $1$ and $5$.

2. What is norm of partition $\{0,3,3.1,3.2,7,10 \}$ of interval $[0,10]$.

- (A) $10$
- (B) $3$
- (C) $3.8$
- (D) $0.1$

(C): Maximum distance between any two points of the partition is norm, which is $7-3.2=3.8$.