Function goes to 0 if all derivatives are $L^2$ Suppose that all derivatives of some function $f$ are $L^2$, so that $\left\lVert\frac{\partial^kf}{\partial x^k}\right\rVert<\infty$ for all $k\ge0$. Then is it true that $f\to0$ as $x\to\infty$?
I think the answer is yes, but I'm having trouble proving this. Any help would be great!

Yes, this follows, for instance, from Morrey's inequality, since this gives $f$ uniformly continuous and square integrable.
Added: If you're interested in one dimension you can actually give an easy proof using the fundamental theorem of Calculus and Hölder's inequality:
$$
|f(x)-f(y)|\leq \int_{y}^x |f'(t)|\, dt \leq |x-y|^{1/2}\| f'\|.
$$
This gives the uniform continuity of $f$.
