====== 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]]. 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) ===== ==== Real Number System==== 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 $(a,b)$, $a\frac{1}{3}$. 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 $2$. ==== Sequence of Numbers ==== 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|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) $|s_n-s_m|\epsilon$. * (C) $|s_n-s_m|<\epsilon$ for all $n,m>n_0$. * (D) $|s_n-s_m|<\epsilon$ for all $n,mSee 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. ==== Series of Numbers ==== 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. ==== Limit of functions ==== 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): $\lim_{x\to 3}f(x)=\frac{x^2-5x+6}{x-3}=\lim_{x\to 3}\frac{(x-2)(x-3)}{x-3}$ $=\lim_{x\to 3}(x-2) = 1$. ==== Riemann Integrals ==== 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 $7-3.2=3.8$.