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 page will be updated periodically.

Short questions

1. What is the difference between rational and irrational numbers?
2. Is there a rational number exists between any two rational numbers.
3. Is there a real number exists between any two real numbers.
4. Is the set of rational numbers is countable?
5. Is the set of real numbers is countable?
6. Give an example of sequence, which is bounded but not convergent.
7. Is every bounded sequence is convergent?
8. Is product of two convergent sequences is convergent?

Multiple choice questions (MCQs)

1. Whis is not true about number zero.
   • (A) Even
   • (B) Positive
   • (C) Additive identity
   • (D) Additive inverse of zero
2. Which one of them is not interval.
   • (A) $(1,2)$
   • (B) $\left(\frac{1}{2},\frac{1}{3}\right)$
   • (C) $\left[3.\pi \right]$
   • (D) $(2\pi ,180)$
3. A number which is neither even nor odd is
   • (A) 0
   • (B) 2
   • (C) $2n$ such that $n\in Z$
   • (D) $2\pi$
4. A number which is neither positive nor negative is
   • (A) 0
   • (B) 1
   • (C) $\pi$
   • (D) None of these
5. Concept of the divisibility only exists in set of …………..
   • (A) natural numbers
   • (B) integers
   • (C) rational numbers
   • (D) real numbers
6. If a real number is not rational then it is ……………
   • (A) integer
   • (B) algebraic number
   • (C) irrational number
   • (D) complex numbers
7. Which of the following numbers is not irrational.
   • (A) $\pi$
   • (B) $\sqrt{2}$ * (C) $\sqrt{3}$
   • (D) 7
8. A set $A$ is said to be countable if there exists a function $f:A\to N$ such that
   • (A) $f$ is bijective
   • (B) $f$ is surjective
   • (C) $f$ is identity map
   • (D) None of these
9. Let $A=\{x|x\in N\wedge x^2\le 7\}$. Then supremum of $A$ is
   • (A) 7
   • (B) 3
   • (C) does not exist
   • (D) 0
10. A convergent sequence has only ……………. limit(s).
    • (A) one
    • (B) two
    • (C) three
    • (D) None of these
11. A sequence $\left\{s_n\right\}$ is said to be bounded if
    • (A) there exists number $\lambda$ such that $|s_n|<\lambda$ for all $n\in Z$.
    • (B) there exists real number $p$ such that $|s_n|<p$ for all $n\in Z$.
    • (C) there exists positive real number $s$ such that $|s_n|<s$ for all $n\in Z$.
    • (D) the term of the sequence lies in a vertical strip of finite width.
12. 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.
13. A sequence $\left\{\right(-1{)}^n\}$ is
    • (A) convergent.
    • (B) unbounded.
    • (C) divergent.
    • (D) bounded.
14. A sequence $\left\{\frac{1}{n}\right\}$ is
    • (A) bounded.
    • (B) unbounded.
    • (C) divergent.
    • (D) None of these.
15. A sequence $\left\{s_n\right\}$ is said be Cauchy if for $ϵ>0$, there exists positive integer $n_0$ such that
    • (A) $|s_n-s_m|<ϵ$ for all $n,m>0$.
    • (B) $|s_n-s_m|<n_0$ for all $n,m>ϵ$.
    • (C) $|s_n-s_m|<ϵ$ for all $n,m>n_0$.
    • (D) $|s_n-s_m|<ϵ$ for all $n,m<n_0$.
16. Every Cauchy sequence has a ……………
    • (A) convergent subsequence.
    • (B) increasing subsequence.
    • (C) decreasing subsequence.
    • (D) positive subsequence.
17. 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.
18. Let $\left\{s_n\right\}$ 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) ${lim}_{n\to \infty }{s}_{n+1}=s^2$.
19. Every convergent sequence has …………….. one limit.
    • (A) at least
    • (B) at most
    • (C) exactly
    • (D) none of these
20. If the sequence is decreasing, then it …………….
    • (A) converges to its infimum.
    • (B) diverges.
    • (C) may converges to its infimum
    • (D) is bounded.
21. If the sequence is increasing, then it …………….
    • (A) converges to its supremum.
    • (B) diverges.
    • (C) may converges to its supremum.
    • (D) is bounded.
    •