Proof of Riemann-Lebesgue lemma: what does "integration by parts in each variable" mean? I was reading the proof of the Riemann-Lebesgue lemma on Wikipedia, and something confused me. It says the following:

What does the author mean by "integration by parts in each variable"? If we integrate by parts with respect to $x$, then (filling in the limits, which I believe are $-\infty$ and $\infty$) I get $$\hat{f}(z) = \left[\frac{-1}{iz}f(x)e^{-izx}\right]_{-\infty}^{\infty} + \frac{1}{iz}\int_{-\infty}^{\infty}e^{-izx}f'(x)dx.$$
I think I am missing something here...it's not clear to me why the limit at $-\infty$ of the first term should exist. Can anyone clarify this for me?
Thanks very much.

This question was answered by @danielFischer in the comments.
