Real Analysis: Short Questions and MCQs

1. What is not true about number zero.

• (A) Even
• (B) Positive
• (C) Additive identity
• (D) Additive inverse of zero
  See Answer

  (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) $(2\pi,180)$
  See Answer

  (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) $2\pi$
  See Answer

  (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
  See Answer

  (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
  See Answer

  (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
  See Answer

  (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
  See Answer

  (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
  See Answer

  (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
  See Answer

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

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

• (A) one
• (B) two
• (C) three
• (D) None of these
  See Answer

  (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.
  See Answer

  (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.
  See Answer

  (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.
  See Answer

  (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.
  See Answer

  (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 $n,m<n_0$.
  See Answer

  (C): Definition of Cauchy sequence.

7. Every Cauchy sequence has a ……………

• (A) convergent subsequence.
• (B) increasing subsequence.
• (C) decreasing subsequence.
• (D) positive subsequence.
  See Answer

  (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.
  See Answer

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

  (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
  See Answer

  (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.
  See Answer

  (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.
  See Answer

  (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
  See Answer

  (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.
  See Answer

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

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.
  See Answer

  (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.
  See Answer

  (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.
  See Answer

  (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
  See Answer

  (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.
  See Answer

  (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
  See Answer

  (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) $p>1$.
  See Answer

  (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
  See Answer

  (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 $\lim a_n=0$.
  See Answer

  (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.
  See Answer

  (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) $\{ a_n \}$
  See Answer

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

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) $|f(x)-L| \geq \varepsilon$
  See Answer

  (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
  See Answer

  (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.
  See Answer

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

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) $\{1,2.1,3,4,5.5 \}$
  See Answer

  (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) $0.1$
  See Answer

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