= Solution
{c}
Use <uniform convergence> to choose $N$ with $|f_n(x)-f_N(x)|\leq1$ for all $n\geq N$ and $x\in E$; this follows, for example, from the <Uniform Cauchy criterion>. Since each $f_j$ is a <bounded function>, put $M_j=\sup_E|f_j|<\infty$ for $1\leq j\leq N$. Then
$$
|f_n(x)|\leq M:=1+\max_{1\leq j\leq N}M_j
$$
for every $n$ and $x$. Thus $(f_n)$ is a <uniformly bounded family of functions>. The finite initial segment is essential: <uniform convergence> directly controls only a <tail of a sequence>.
Back to article page
