$\{f_n\}$ is a family of functions continuous on the interval $(0,1)$. I'm forgetting how to define what it means for $\{f_n\}$ to be bounded and my textbook/google searches aren't providing what I need.
Is it $$\sup_{0< x< 1}|f_n(x)|\leq K$$ for some $K$ and for all $n$?
It is ambiguous. There are several interpretations:
Pointwise bounded: For all $x\in (0,1)$, $\sup_n |f_n(x)| < \infty$ .
Uniformly bounded: $\sup_n \|f_n\|_\infty < \infty$ where $\|f_n\|_\infty := \sup_{x \in (0,1)} |f(x)|$.
Entrywise bounded: For all $n$, $f_n$ is bounded.
I guess the second interpretation is the most likely, but really one cannot tell without context.