Here is one application, which may not seem spectacular to the modern mathematician, but it has many profound applications.

Suppose that $X$ is a reflexive Banach space $E: X\to (-\infty, \infty]$ a convex function such that

$$ \lim_{\Vert x\Vert\to\infty} E(X)=\infty, $$

and

$$ E(x)\leq \liminf_{y\to x} E(x). \;\;\forall x\in X. $$

Then there exists $x_0\in X$ such that

$$ E(x_0)\leq E(x),\;\;\forall x\in X. $$

For example, one can use this to settle the so called Dirichlet principle which generated many debates in the 19th century.