Question

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

Accepted Answer

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

|fn(x)| ≤ Pn for x ∈ B.

Let N be such that

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

So, for n ≥ N and x ∈ B,

|fn(x)|< |fn(x) − f(x)| + |f(x) − fN (x)| + |fN(x)| < 2 + PN .

Let

P = max{P1, · · · , PN-1, 2 + Pn}.

So, for any x ∈ B,

|fn(x)| ≤ P, n = 1, 2, 3, . . . .

Hence, {fn} is uniformly bounded.

Question #115372

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