Arzelà-Ascoli theorem

For a compact Hausdorff space $K$, a subset of $C(K)$ is relatively compact in the uniform norm exactly when it is equicontinuous and pointwise bounded. On a compact metric space, this says that every uniformly bounded equicontinuous sequence has a uniformly convergent subsequence.

If a thrice differentiable solution has both endpoint values and both endpoint first derivatives bounded in modulus by $N$, it obeys $|f|\le N$ and $|f^{\prime }|\le N$ throughout the interval. A positive interior maximum of $f$ would give $f^{\prime \prime }=f>0$, and a negative interior minimum would give $f^{\prime \prime }=f<0$, both impossible. For $w=f^{\prime }$, differentiation gives $w^{\prime \prime }=w+2w{w}^{\prime }$, so the same argument applies at an extremum of $w$. The resulting uniform bound and Lipschitz constant imply total boundedness of the solution family by the Arzelà-Ascoli theorem.

A subset of a topological space is relatively compact when its closure is compact.

If successive subsequences converge uniformly on an exhaustion $K_1\subset K_2\subset \cdots$, the diagonal sequence converges uniformly on every fixed $K_m$, hence locally uniformly on their union.

For a compactly supported nonzero function $\psi$, the translates $f_n(x)=\psi (x-n)$ converge uniformly to zero on every compact subset of $\mathbb{R}$, while $\Vert f_n{\Vert }_{\infty }=\Vert \psi {\Vert }_{\infty }$ prevents uniform convergence on all of $\mathbb{R}$.