Question #115372
If fn converges to f and FN is bounded on a set S prove that {fn} is uniformly bounded on S
Expert's 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.
