1) let (f_{n}) be a sequence of real- valued functions that converges uniformly to f_{*} on a set a\subseteq r , where each f_{n} is bounded on a, i. e there exists mn> 0 such that |f_{n}(x)|\leq m_{n} for all x\in a .
a) show that f_{*} bounded on a.
b)show that there exists m> 0 shuch that |f_{n}(x)|\leq m for all x\in a and all n.
c) let (a_{n}) be a bounded real sequence , and denote by a_{n}f_{n} the product of a_{n} and f_{n} . show that (a_{n}f_{n}) has a subsequence which converges uniformly on a.