In the book by David R. Adams/Lars Inge Hedberg, Function Spaces and Potential Theory, page 29, there is a line saying something like the following:
Suppose that $\varphi_{n}\geq 0$ is continuous and $\varphi_{n}(x)\uparrow\varphi(x)$ for all $x\in\mathbb{R}^{N}$ for $n\rightarrow\infty$. If $\varphi(x)>1$ for all $x\in K$, where $K$ is compact, then by lower semi-continuity, there are $\delta>0$ and $n$ such that $\varphi_{n}(x)\geq 1+\delta$ for all $x\in K$.
I have my own reasoning to argue the existence of such $\delta>0$ and $n$, but it seems to me that the property that $\varphi=\sup_{n}\varphi_{n}$ being lower semi-continuity is not used anywhere.
Therefore I ask for proof verification if I did missing something in my following argument:
For each $x\in K$, let $\delta_{x}>0$ be such that $\varphi(x)>1+\delta_{x}$. Since $\varphi_{n}(x)\uparrow\varphi(x)$ pointwise, then there is an $n_{x}$ such that $\varphi_{n_{x}}(x)>1+\delta_{x}$.
The continuity of $\varphi_{n_{x}}$ implies that some open set $V_{x}$ is such that \begin{align*} \varphi_{n_{x}}(y)>1+\delta_{x}/2,~~~~y\in V_{x}. \end{align*} Now the open sets $V_{x}$ cover the compact set $K$, it reduces to a finite covering that \begin{align*} K\subseteq V_{x_{1}}\cup\cdots\cup V_{x_{n_{0}}} \end{align*} for $x_{1},...,x_{n_{0}}\in K$.
We let $n=\max(n_{x_{1}},...,n_{x_{n_{0}}})$ and $\delta=\min(\delta_{x_{1}},...,\delta_{x_{n_{0}}})$.
Now let $z\in K$ so $z\in V_{x_{i}}$ for some $i=1,...,n_{0}$ . Hence \begin{align*} \varphi_{n}(z)\geq\varphi_{x_{i}}(z)>1+\delta_{x_{i}}/2\geq 1+\delta/2. \end{align*}
Is there any flaw in my argument?
Your proof is correct.
But you can also argue as follwos:
The continuity of $\varphi_{n_{x}}$ implies that some open set $V_{x}$ is such that \begin{align*} \varphi_{n_{x}}(y)>1+\delta_{x},~~~~y\in V_{x}. \end{align*}
etc.
However, you do not use the continuity of the $\varphi_n$ in full extent. The argument that $\varphi_n(x)>r$ implies that $\varphi_n(y)>r$ for $y$ in some neighborhood of $x$ is based on the lower semi-continuity of the $\varphi_n$ .