I search a reference (book/paper) for the following result (or generalizations).
Let $K$ be a compact topological space and $(f_\alpha)$ be a net of real continuous functions on $K$. We suppose that $(f_\alpha)$ is uniformly convergent to $f$. Let $m$ the minimum of $f$ over $K$ and $m_\alpha$ the minimum of $f_\alpha$. Then $$ \lim_{\alpha} m_\alpha=m. $$