Corollary

Let $(u_n)$ be an increasing (resp. decreasing) sequence of continuous real-valued functions on $X$. If the upper (resp. lower) envelope $f$ of the sequence $(u_n)$ is finite and continuous on $X$, then the sequence $(u_n)$ converges uniformly to $f$ in $X$.

It is clear that the conclusion of Theorem 1 is no longer necessarily valid if $X$ is no longer assumed to be compact, as is shown by the example of the decreasing sequence of functions $x/(n + x)$ in $\mathbf{R}_+$.