3. For each n€ Z+ let fn : [0, 1] → R, fn(x) = 1/(1+x"). (a) Prove that (fn) converges uniformly. 1/2 (b) Compute the limit lim n→X S" fn, rigorously justifying your answer.

Suppose {f} is a sequence of functions, differentiable on [a, b] and such that {fn (xo)} converges for some point xo on [a, b]. If {f'n} converges uniformly on [a, b], then {f} converges uniformly on [a, b], to a function f, and (a ≤ x ≤ b). f'(x) = lim f'(x) n2-00 Can we replace [a, b] by [0, ∞o)? Why?

Let YN == √N+1' - N' for NEn. Show that Find the limit. (UNY) cri Converges.

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.