Elements of Real Analysis — DU B.A. (Prog) Mathematics as Major Previous Year Question Papers

3. Part of the questions to be attempted together. (a) Define infimum and supremum of a set. Find the infimum and supremum, if exist, of the following sets : (b) Let F be an Archimedean ordered field, A s F, and u e F. Prove that = sup A if and only if for all 6 > 0, (i) for all x e A, x < u + e, and (ii) there exists x e A, such that x> - e. (c) Show that in any ordered field F, 2. (a) Suppose A c B are subsets of a complete ordered field F and A # $. Prove that (i) If B is bounded below, then so is A, and inf A ≥ inf B. (i) If B is bounded above, then so is A, and sup A ≤ sup B. (b) State Completeness property of any ordered field F. Prove that any complete ordered field is (c) For any a eR, with lal <1, show that lima" = 0. 3. (a) Define the convergence of a sequence of real numbers. Use the definition of convergence to prove that 1m (-3 = 3 (b) If (a), (b) and (c,) are sequences such that lim a lim an =1, limon = l, and Vn e N, a, 5b, 5 Ca then prove that limbn= 1. (c) Use the algebra of limits to find the following limits-n°+3n+5 4. (a) Prove that if {a) is a bounded sequence and (br) converges to zero, then {a_b} → 0. Use this result to prove that lim Cosn = 0. (b) Consider the sequence (x} defined inductively by x, = 1 and V ne N, N, =VXn+ 2. Prove that {x) converges and find its limit. (c) Show that every Cauchy sequence of real numbers is a bounded sequence. Is the converse true? 5. (a) Prove that if the series Za, converges, then an →O. Is the converse true? Justify your answer. (b) Test for the convergence of following series: (c) Test for the convergence of the series 6. (a) State the Ratio test for convergence of a series. Determine if the series Z0=1 en converges or diverges. (b) Use Raabe's test to determine if the series with (c) Determine if the alternating series Ens, (=1)4t2 cos n n2 converges absolutely. Hence or otherwise test if the series converges conditionally.