RGU Question Paper Mathematics Semester IV 2021: Real Analysis-I (New)

 June-2021

B.A/B. Sc. (IV Semester) Examination

MATHEMATICS

Paper: MATH-241 (NEW)

(Real Analysis-I)

Full Marks : 80

Pass Marks : 35%

Time : Three Hours

Notes: (i) Answer all questions.

(ii) The figures in the margin indicate full marks for the questions.

I. Answer any five parts: 𝟐 × 𝟓 = 𝟏𝟎

(a) Show that the set 𝑆 = {𝑥: 0 < 𝑥 < 1, 𝑥 ∈ ℝ} is open but not closed.

(b) Show that the set of rational numbers in [0,1] is countable.

(c) Determine whether the limit of the given function exists (or) not if

(d) Discuss the continuity of the function [𝑥] at 𝑥 = 3.

(e) Give an example to show that continuity is not a sufficient condition for derivability of a function.

(f) Give an example of a Cauchy sequence which is not convergent.

(g) Show that the series $\frac{1}{2}+\frac{2}{3}+\frac{3}{4}+...$ is not convergent.

II. Answer any four parts: 𝟓 × 𝟒 = 𝟐𝟎

(a) State and prove Bolzano Weierstrass theorem for sets.

(b) What do you mean by discontinuous function? Discuss the types of discontinuities with suitable examples.

(c) Evaluate 

(d) State and prove Cauchy’s general principle of convergence for series.

(e) Show that the series

converges for 𝑥 ≤ 1 and diverges for 𝑥 > 1.

(f) Prove that every bounded sequence with a unique limit point is convergent.

(g) Prove that the sequence $\left\{r_n\right\}$ converges to zero if and only if |𝑟| < 1.

III. Answer any five parts: 𝟏𝟎 × 𝟓 = 𝟓𝟎

(a) (i) Prove that a necessary and sufficient condition for the convergence of a monotonic sequence is that it is bounded.

(ii) Show that the sequence $\left\{S_n\right\}$, where

is convergent. (5+5)

(b)(i) If {an} is any sequence, then prove that

(ii) If $\left\{a_n\right\}$ is any sequence such that $\underset{n\to \infty }{\lim }\frac{a_{n+1}}{a_n}=l$, where |𝑙| < 1,

then show that $\underset{n\to \infty }{\lim }{a}_n=0$. (5+5)

(c)(i) Prove that a set is closed if and only if its complement is open.

(ii) Prove that every open set is a union of open intervals.

(5+5)

(d) Prove that a positive term series $\sum \frac{1}{n^p}$ is convergent iff 𝑝 > 1.

(e) (i) State and prove Lagrange’s mean value theorem.

(ii) Show that

𝑖𝑓 0 < 𝑢 < 𝑣 and deduce that

(5+5)

(f) Test the convergence of the series

(g) (i) Prove that a function which is continuous on a closed interval is also uniformly continuous on that interval.

(ii) Show that the function 𝑓(𝑥) = 1/𝑥 is not uniformly continuous on (0,1]. (5+5)