Behavior of a sequence of uniformly continuous functions at the boundary of the interval

Question

Consider a sequence of uniformly continuous functions on an interval $f_n : [a,b] \rightarrow \mathbb{R}$ that uniformly converge to a uniformly continuous function $f$. Assume f(a) takes an indeterminate form, but $f(x)$ is well defined and uniformly continuous over $(a,b]$.

We can define an extension function $F$ such that $F(x) = f(x)$ for $x \in (a,b]$, and $F(a) = \lim_{x\rightarrow a} f(x)$.

Prove that the sequence ${f_n(a)}$ converges to $F(a)$.

My attempt at the proof:

Take any number $\epsilon>0$. We need to find $M$ such that when $n>M$, $|f_n(a)-F(a)| < \epsilon$

Select $c$ such that for all $\delta \in (a,c)$, $sup |f_n(\delta) - f_n(a)| < \frac{\epsilon}{3}$ and $ |F(a)-f(\delta)| < \frac{\epsilon}{3}$. I think because all $f_n$ are continuous we should be able to achieve this. Is that reasonable? Note: the functions $f_n$ may not be monotonously increasingly or decreasing from $a$ to $c$ but they are bounded because they are continuous. I am a little unsure about this part.

Now select $M$ such that for all $n> M$ , $|f_n(\delta) - f(\delta)| < \frac{\epsilon}{3}$. This is possible because $f_n(x)$ converges to $f(x)$

Now when $n > M$ $$|f_n(a)-F(a)| = |f_n(a) - f_n(\delta) + f_n(\delta) - f(\delta) + f(\delta) - F(a)|$$

$$|f_n(a)-F(a)| \le |f_n(a) - f_n(\delta)| + | f_n(\delta) - f(\delta)| + | f(\delta) - F(a)|$$

$$|f_n(a)-F(a)| \le \epsilon$$

Does this work? Is there a theorem that already takes care of this? Thank you in advance!

Answer 1

Your proof looks good. Just a few pointers:

1. You didn't specify but I believe you mean that $f_n \to f$ uniformly, not pointwise (but pointwise convergence of uniformly continuous functions over a compact interval is uniform).
2. "Select $c$ such that for all $\delta \in (a,c)$, $\sup|f_n(\delta) - f_n(a)| < \frac{\epsilon}{3}$ and $ |F(a)-f(\delta)| < \frac{\epsilon}{3}$." Indeed this is valid, but the functions being continuous alone is not enough. You require uniform continuity since $\delta$ is arbitrary in the interval $(a, c)$.