# Real Analysis: Short Questions and MCQs

We are going to add short questions and MCQs for Real Analysis. The subject is similar to calculus but little bit more abstract. This is a compulsory subject in MSc and BS Mathematics in most of the universities of Pakistan. The author of this page is Dr. Atiq ur Rehman. The page will be updated periodically.

## 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?

## Multiple choice questions (MCQs)

- Whis is not true about number zero.
- (A) Even
- (B) Positive
- (C) Additive identity
- (D) Additive inverse of zero

- Which one of them is not interval.
- (A) $(1,2)$
- (B) $\left(\frac{1}{2},\frac{1}{3} \right)$
- (C) $[3. \pi]$
- (D) $(2\pi,180)$

- A number which is neither even nor odd is
- (A) 0
- (B) 2
- (C) $2n$ such that $n \in \mathbb{Z}$
- (D) $2\pi$

- A number which is neither positive nor negative is
- (A) 0
- (B) 1
- (C) $\pi$
- (D) None of these

- Concept of the divisibility only exists in set of …………..
- (A) natural numbers
- (B) integers
- (C) rational numbers
- (D) real numbers

- If a real number is not rational then it is ……………
- (A) integer
- (B) algebraic number
- (C) irrational number
- (D) complex numbers

- Which of the following numbers is not irrational.
- (A) $\pi$
- (B) $\sqrt{2}$
- (C) $\sqrt{3}$
- (D) 7

- 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

- Let $A=\{x| x\in \mathbb{N} \wedge x^2 \leq 7 \}$. Then supremum of $A$ is
- (A) 7
- (B) 3
- (C) does not exist
- (D) 0

- A convergent sequence has only ……………. limit(s).
- (A) one
- (B) two
- (C) three
- (D) None of these

- 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.

- 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.

- A sequence $\{(-1)^n\}$ is
- (A) convergent.
- (B) unbounded.
- (C) divergent.
- (D) bounded.

- A sequence $\{\frac{1}{n} \}$ is
- (A) bounded.
- (B) unbounded.
- (C) divergent.
- (D) None of these.

- 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 $n,m<n_0$.

- Every Cauchy sequence has a ……………
- (A) convergent subsequence.
- (B) increasing subsequence.
- (C) decreasing subsequence.
- (D) positive subsequence.

- 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.

- 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) $\lim_{n\to\infty}s_{n+1}=s^2$.

- Every convergent sequence has …………….. one limit.
- (A) at least
- (B) at most
- (C) exactly
- (D) none of these

- If the sequence is decreasing, then it …………….
- (A) converges to its infimum.
- (B) diverges.
- (C) may converges to its infimum
- (D) is bounded.

- If the sequence is increasing, then it …………….
- (A) converges to its supremum.
- (B) diverges.
- (C) may converges to its supremum.
- (D) is bounded.

- If a sequence converges to $s$, then ………….. of its sub-sequences converges to $s$.
- (A) each
- (B) one
- (C) few
- (D) none

- 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.

- 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) $\{ a_n \}$

- Let $\sum a_n$ be a series of non-negative terms. Then it is convergent if ………..
- (A) it is bounded.
- (B) it may bounded.
- (C) it is unbounded.
- (D) it may unbounded.

- 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

- A series $\sum \frac{1}{n^p}$ is convergent if
- (A) $p\geq 1$.
- (B) $p\leq 1$.
- (C) $p>1$.
- (D) $p<1$.

- 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

- 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 $\lim a_n=0$.

- 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.

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