I am a little insecure about my proof because the book provide a longer one, can you check it? Thanks in advance
Let $I\subseteq \mathbb{R}$ be an interval and $f,f_n$ functions from I to $\mathbb{R}$ for $n\in\mathbb{N}$ such that $(f_n)$ uniformly converges to $f$. Let $t_0 \in I$ such that for every $n$ exists $l_n=\lim_{t\to t_0}f_n(t)$. Prove that $(l_n)$ converges.
Let $\epsilon >0$ and $n,m \in \mathbb{N}$; we have $|l_n-l_m|=|lim_{t \to t_0}(f_n(t)-f_m(t))|=lim_{t \to t_0}|f_n(t)-f_m(t)| \le lim_{t \to t_0}\epsilon=\epsilon$ if $n, m$ are sufficiently large because $(f_n)$ is uniformly convergent. So $(l_n)$ is a Cauchy sequence and therefore it converges.