I was given the following question as part of my Calculus 2 assignment: Let $\{f_n\}$ be a sequence of functions that converge to $f$, but doesn't converge uniformly, over an interval $I$. Given $p\in I$ show that $f_n\underset{n\to\infty}{\longrightarrow} f$ (not uniformly) over the interval $I\setminus\{p\}$.
Here is what I started writing: We know that $f_n\longrightarrow f$ but not uniformly, therefore there exists $L>0$ such that $\sup_{x\in I}|f_n(x)-f(x)|\longrightarrow L$. We'll denote the supremum with the sequence $\{a_n\}$ meaning $\forall n\in\mathbb{N}\colon\sup_{x\in I}|f_n(x)-f(x)|=|f_n(a_n)-f(a_n)|$.
I have no idea really how to articulate what I want to say. Intuitively I know that, for an infinite set of cardinality $\aleph$, removing a finite number of elements does not change the supremum of the set. I am just having a problem explaining that rigorously and I would like a hint or a guide in the right direction on how to write that thought.