1. Let (n)-1 be a sequence of bounded real-valued functions on X. (a) If ƒn ⇒ ƒ on X, show that ƒ is bounded on X. (b) If (fn) 1 converges pointwise to a bounded function f on X, must the convergence be uniform? Justify.

Let                          fn(x) = nx/1 + nx2  (d) Is the convergence uniform on (1,∞)?

Let (fn) be a sequence of functions that each fn is continuous [0, 1] and differentiable on (0, 1). Suppose (fn) converges uniformly to a function f on [0, 1]. Which of the following statement (s) must be true? (i) of = lim, So fn. (ii) f is uniformly continuous on [0, 1] (iii) f is differentiable on (0, 1).

only (c) and don't use AI