I know if a real valued function $g$ is continuous on a closed and bounded interval $[a,b]$, then it is bounded. However, I am not sure whether the following holds for sequences of functions:
$\textbf{Q1}:$ Consider a sequence of real valued functions $g_{n}$ on $[a,b]$. Does it follow that {$g_{n}$} is point-wise bounded?
$\textbf{Q2}:$ If we assume further that {$g_{n}$} converges uniformly to $g$. Do we get uniform boundedness of {$g_{n}$} on $[a,b]$?
is definitely false. Just take $f_n(x) = nx$ restricted to $[a,b]$.
is not necessarily true since, as is the case with sequences of numbers, you don't have to worry about the head of the sequence. All sorts of things can happen before you start converging. However, for infinitely many terms you'll need $g_n$ to be uniformly bounded otherwise you won't even get pointwise convergence to a continuous $g$.