Question

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

Answer 1

We need to assume that $f_n\to f$ uniformly on $S$. Hence, $\exist\, N\in\mathbb N \, \forall m,n\geqslant N \colon |f_n(x)-f_m(x)|<1$. Also $\exist\, C_1,C_2,\dots>0 \colon |f_n(x)|\leqslant C_n,\, x\in S,\,n\in\mathbb N.$ So, for $n\geqslant N,\, x\in S$ we have $|f_n(x)|\leqslant|f_n(x)-f_N(x)|+|f_N(x)|<1+C_N$. Taking $C=\max\{C_1,\dots,C_N\}+1$ we have $\forall n\in\mathbb N\colon |f_n(x)|\leqslant C$ because $\max\{C_1,\dots,C_N\}$ bounds $|f_n|$ for $n<N$.