1. (a) Use Bernoulli's inequality to prove that (1+ (b) Prove that Vn< (1+ 1 VT) ² (c) Prove that lim (√√/n) = 1. n √₁)" = for all n € N. > √n for all n € N.

solve using epsilon deltas

2. Let x₁>1 and Xn+1 = 2 - 1/xn for n E N. show that (xn) is bounded and monotone. Find the limit. 3 Let x. >2 and r. for n N Show that (x..) is decreasing and bounded

Let x1 = 2 and let xn+1 = 1/2 *xn + 1/xn. a. Prove that xn is decreasing and bounded from below. Hint: First prove that x^2 n+1 − 2 > 0. b. Determine limn→∞xn.