Elementary Real Analysis — DU B.Sc. (Hons.) Mathematics Previous Year Question Papers

3. The parts of each question are to be attempted together. RY 1. (a) Find all x € R that satisfy both |2x - 3| < 5 and |x + 1| > 2 simultaneously. (b) Let U = (x E R: 0 < x < 1). Show that each element of U has some e -neighbourhood of it contained in U. (c) Let S= (x € I: x> 0). Does S have lower bounds? Does S have upper bounds? Does inf S exist? Does sup S exist? Prove your statements. (d) Define supremum of a non-empty subset S of R. Show with an example that set can have a supremum without having a maximum. Justify your statements. 2. (a) Is the set S = (1 - -: n € N) bounded? Find its supremum and infimum if they exist. Justify your statements. (b) Let A and B be two subsets of R which are both bounded above. Show that A UB is a bounded above set. Find sup (AUB) in terms of sup A and sup B. Prove your statements. (C) Suppose that A and B are non-empty bounded subsets of R that satisfy a ≤ b for all a E A and all be B. Prove that supA ≤ inf B. (d) Show that a sequence in R can have at most one limit. (a) Show that lim c1/n= 1 forc>0. (b) If (an) and (bn) converge to a and b respectively, prove that (ann) converges to ab. (с) State and prove the Squeeze Theorem. (d) Establish the convergence or the divergence of the sequence (Yn), where Уп = +•••+ for nEN n +1 n + 2 2n 4. (a) Prove that if (*n) is a monotone increasing and bounded above sequence of real numbers, then it is convergent. (b) IF Lo Xn =x, for any real sequence (x). Then show that Lim xl = |xl. Give an example to show that the convergence of (IXnl) need not imply convergence of (x). (c) Find limit inferior and limit superior of the following sequences: (d) State Cauchy Convergence Criterion for sequencos. Hence show that the sequence (an), defined by an = 1++.+- •, does not converge. 5. (a) Prove that for a 0, the geometric series En-o ar" = a + ar + ar? + ar3 +... is convergent if and only if Irl < 1. (b) Find the sum of the series n=1 (c) Suppose 2 an and [br are non negative series. Let lim an = L. If 0 < L m, then prove that either both series converge or hoth series diverge. (d) Check the convergence of the following series: 6. (a) Prove that the series Prove that the series 2 near ny conces and aily ip > 1. (b) State and prove the alternating series test. (c) Define an alternating series. Check the convergence of the series (d) Determine for each of the following series whether the series converges absolutely, converges conditionally or diverges. (i) (-1)n 1 (11) (-1)n COS TATE