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$\lim I[\phi_{n}] \geq I[u]$
no matter what sequence $\phi_{n}$ tending to $u$ is considered.
$\underline{Compactness\ in\ Function\ Space,\ Arzela's\ Theorem\ and\ Applications}$

In the ordinary theory of clear maxima or minima, the existence of a greatest or smallest value of a function in a closed domain is assured by the Bolzano-Weierstrass convergence theorem: a bounded set of points always contains a convergent subsequence. This fact, together with the continuity of the function, serves to secure the existence of an extreme value.
As seen before, in the calculus of variations the continuity of the function space often has to be replaced by a weaker property, semi-continuity; for, when $\phi_{n}$ converges to $u$, generally $I(\phi_{n}) \ne I(u)$. Another difficulty in the calculus of variations arises from the fact that the Bolzano-Weierstrass convergence theorem does not hold if the elements of the set are no longer points on a line or in a n-dimensional space, but functions, curves or surfaces.
As an example, let us consider a $\phi$ with the set of functions
$$\int_0^1 e^{xy} g(y) dy = f(x)$$ where $g(y)$ is any piecewise continuous function such that $|g(y)| \lt 1$ and where $x$ has an upper bound. Then $$|f(x) - f(\zeta)| \leq \int_0^1 |g(y)| |e^{xy} - e^{\zeta y}| dy$$

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