# 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, PhD. 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?
- 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)

- 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

- 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.

- 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): $2\pi$ is not an integer.

- 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 .

- 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

- 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.

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

(D): Its easy to see

- 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.

- 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.

- 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 $0<|x-c|<\delta$.
- $|f(x)-L| > \varepsilon$
- $|f(x)-L| < \varepsilon$
- $|f(x)-L| \leq \varepsilon$
- $|f(x)-L| \geq \varepsilon$

- If $\lim_{x \to c}f(x)=L$, then ………… sequence $\{x_n\}$ such that $x_n \to c$, when $n\to \infty$, one has $\lim_{n \to \infty}f(x_n)=L$.
- for some
- for every
- for few
- none

- Let $f(x)=\frac{x^2-5x+6}{x-3}$, then $\lim_{x\to 3}f(x)=$………..
- $-1$
- $0$
- $1$
- doesn't exist.

- Which one is not partition of interval $[1,5]$.
- $\{1,2,3,5 \}$
- $\{1,3,3.5,5 \}$
- $\{1,1.1,5 \}$
- $\{1,2,3,4,5 \}$

- What is norm of partition $\{0,3,3.1,3.2,7,10 \}$ of interval $[0,10]$.
- $10$
- $3$
- $3.8$
- $0.1$