A sequence of functions fn is said to be uniformly bounded on a set E if and only if there is an M > 0 such that |fn (x) < M for all x ∈ E and all n ∈ N.
Suppose that for each n ∈ N, fn: E → R is bounded. If fn → f uniformly on E, as n → N, prove that {fn} is uniformly bounded on E and f is a bounded function on E.