Real Analysis: Short Questions and MCQs

<callout type=“info” icon=“true”> 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. </callout>

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 countable?
5. Is the set of real numbers countable?
6. Give an example of sequence, which is bounded but not convergent.
7. Is every bounded sequence convergent?
8. Is product of two convergent sequences convergent?
9. Give an examples of two divergence sequences, whose sum is convergent.
10. Prove that $\left\{\frac{1}{n+1} \right\}$ is decreasing sequence.
11. Is the sequence $\left\{\frac{n+2}{n+1} \right\}$ is increasing or decreasing?
12. If the sequence $\{x_n\}$ converges to 5 and $\{y_n\}$ converges to 2. Then find $\lim_{n\to\infty z_n}$, where $z_n=x_n-2y_n$.
13. If the sequence $\{x_n\}$ converges to 3 and $\{y_n\}$ converges to 4. Then find $\lim_{n\to\infty z_n}$, where $x_n=2y_n-3z_n$.
14. Give an example to prove that bounded sequence may not convergent.
15. Prove that every convergent sequence is bounded.

<panel> 1. What is not true about number zero.

• (A) Even
• (B) Positive
• (C) Additive identity
• (D) Additive inverse of zero
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  (B): zero is neither positive not negative

</panel> <panel> 2. Which one of them is not interval.

• (A) $(1,2)$
• (B) $\left(\frac{1}{2},\frac{1}{3} \right)$
• (C) $[3. \pi]$
• (D) $(2\pi,180)$
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  (B): In interval $(a,b)$, $a<b$ but $\frac{1}{2}>\frac{1}{3}$.

</panel> <panel> 3. A number which is neither even nor odd is

• (A) 0
• (B) 2
• (C) $2n$ such that $n \in \mathbb{Z}$
• (D) $2\pi$
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  (D): Integers can only be even or odd but $2\pi$ is not an integer.

</panel> <panel> 4. A number which is neither positive nor negative is

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

</panel><panel> 5. Concept of the divisibility only exists in set of …………..

• (A) natural numbers
• (B) integers
• (C) rational numbers
• (D) real numbers
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  (B): In integers, we define divisibility rugosely

</panel><panel> 6. If a real number is not rational then it is ……………

• (A) integer
• (B) algebraic number
• (C) irrational number
• (D) complex numbers
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  (C): Real numbers can be partitioned into rational and irrational.

</panel><panel> 7. Which of the following numbers is not irrational.

• (A) $\pi$
• (B) $\sqrt{2}$
• (C) $\sqrt{3}$
• (D) 7
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  (D): Its easy to see

</panel><panel> 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
  <btn type=“link” collapse=“a8”>See Answer</btn>

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

</panel><panel> 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
  <btn type=“link” collapse=“a9”>See Answer</btn>

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

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

• (A) one
• (B) two
• (C) three
• (D) None of these
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  (A): limit of the sequence, if it exist, is unique.

</panel><panel> 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.
  <btn type=“link” collapse=“a11”>See Answer</btn>

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

</panel><panel> 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.
  <btn type=“link” collapse=“a12”>See Answer</btn>

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

</panel><panel> 4. A sequence $\{(-1)^n\}$ is

• (A) convergent.
• (B) unbounded.
• (C) divergent.
• (D) bounded.
  <btn type=“link” collapse=“a13”>See Answer</btn>

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

</panel> <panel> 5. A sequence $\left\{\dfrac{1}{n} \right\}$ is

• (A) bounded.
• (B) unbounded.
• (C) divergent.
• (D) None of these.
  <btn type=“link” collapse=“a14”>See Answer</btn>

  (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 $n \in \mathbb{N}$.

</panel> <panel> 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 $n,m<n_0$.
  <btn type=“link” collapse=“a15”>See Answer</btn>

  (C): Definition of Cauchy sequence.

</panel><panel> 7. Every Cauchy sequence has a ……………

• (A) convergent subsequence.
• (B) increasing subsequence.
• (C) decreasing subsequence.
• (D) positive subsequence.
  <btn type=“link” collapse=“a16”>See Answer</btn>

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

</panel><panel> 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.
  <btn type=“link” collapse=“a17”>See Answer</btn>

  (B): Cauchy criterion for convergence of sequences.

</panel><panel> 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) $\lim_{n\to\infty}s_{n+1}=s^2$.
  <btn type=“link” collapse=“a18”>See Answer</btn>

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

</panel><panel> 10. Every convergent sequence has …………….. one limit.

• (A) at least
• (B) at most
• (C) exactly
• (D) none of these
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  (C): Every convergent sequence has unique limit.

</panel><panel> 11. If the sequence is decreasing, then it …………….

• (A) converges to its infimum.
• (B) diverges.
• (C) may converges to its infimum
• (D) is bounded.
  <btn type=“link” collapse=“211”>See Answer</btn>

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

</panel><panel> 12. If the sequence is increasing, then it …………….

• (A) converges to its supremum.
• (B) diverges.
• (C) may converges to its supremum.
• (D) is bounded.
  <btn type=“link” collapse=“212”>See Answer</btn>

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

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

• (A) each
• (B) one
• (C) few
• (D) none
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  (A): Every subsequence of convergent sequence converges to the same limit.

</panel><panel> 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.
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  (D): Every subsequence of convergent sequence converges to the same limit.

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<panel> 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.
  <btn type=“link” collapse=“301”>See Answer</btn>

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

</panel><panel> 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.
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  (A)

</panel> <panel> 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.
  <btn type=“link” collapse=“303”>See Answer</btn>

  (C): It is called divergent test

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

• (A) convergent.
• (B) divergent.
• (C) constant.
• (D) none of these
  <btn type=“link” collapse=“304”>See Answer</btn>

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

</panel><panel> 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.
  <btn type=“link” collapse=“305”>See Answer</btn>

  (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.

</panel><panel> 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
  <btn type=“link” collapse=“306”>See Answer</btn>

  (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.

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

• (A) $p\leq 1$.
• (B) $p\geq 1$.
• (C) $p<1$.
• (D) $p>1$.
  <btn type=“link” collapse=“307”>See Answer</btn>

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

</panel><panel> 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
  <btn type=“link” collapse=“308”>See Answer</btn>

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

</panel><panel> 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 $\lim a_n=0$.
  <btn type=“link” collapse=“309”>See Answer</btn>

  (B): Its called alternating series test.

</panel><panel> 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.
  <btn type=“link” collapse=“310”>See Answer</btn>

  (C): It is definition of absolutely convergent.

</panel><panel> 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) $\{ a_n \}$
  <btn type=“link” collapse=“311”>See Answer</btn>

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

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<panel> 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) $|f(x)-L| \geq \varepsilon$
  <btn type=“link” collapse=“401”>See Answer</btn>

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

</panel><panel> 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
  <btn type=“link” collapse=“402”>See Answer</btn>

  (B)

</panel><panel> 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.
  <btn type=“link” collapse=“403”>See Answer</btn>

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

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<panel> 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) $\{1,2.1,3,4,5.5 \}$
  <btn type=“link” collapse=“601”>See Answer</btn>

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

</panel><panel> 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) $0.1$
  <btn type=“link” collapse=“602”>See Answer</btn>

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

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