Let ${f_n}$ be a sequence of bounded functions $f_n:S\to \mathbb R$ that converge uniformly to $f:S\to \mathbb R$. Prove that $f$ is bounded.
Does this prove the statement and are there any flaws in this proof?
Since $f_n$ converges uniformly to $f$ on $S$ , then given $ε=1$ , there exists a positive integer $n_0$ such that as $n≥n_0$ , we have $|f_n(x)-f(x)|≤1$ for all $x∈S$ . Hence , $f(x)$ is bounded on $S$ by the following $|f(x)|≤|f_{n_0}(x)|+1≤Mn_0+1$ for all $x∈S$ where $|f_{n_0}(x)|≤Mn_0$ for all $x∈S$.
Your proof is correct, but it'll be clearer if you say that by the triangle inequality, $|f(x)| \le |f(x) - f_{n_0}(x)| + |f_{n_0}(x)| < 1 + M_{n_0}$ for all $x\in S$.