Answer to Question #115372 in Real Analysis for Sheela John

Question

If fn converges to f and FN is bounded on a set S prove that {fn} is uniformly bounded on S

Answer 1

Suppose that f_n → f uniformly on some set B and for each n, there exists P_n such that

|f_n(x)| ≤ P_n for x ∈ B.

Let N be such that

|f_n(x) − f(x)| < 1, when n ≥ N for x ∈ B.

So, for n ≥ N and x ∈ B,

|f_n(x)|< |f_n(x) − f(x)| + |f(x) − f_N (x)| + |f_N(x)| < 2 + P_N .

Let

P = max{P₁, · · · , P_N-1, 2 + P_n}.

So, for any x ∈ B,

|f_n(x)| ≤ P, n = 1, 2, 3, . . . .

Hence, {f_n} is uniformly bounded.