I've been struggling with a weird complex analysis statement for a few days without making really any headway; I'd really appreciate a hint to get me past where I'm at. The theorem goes like this:
If $f$ is analytic in a connected open set $U$ containing $0$, such that for $n$ large enough we have $|f(\frac{1}{n})|$ $< e^{-n}$, then $f$ is identically zero on $U$.
The book says it uses only basic knowledge about analytic functions; the problem I run into is that the method I try to use, namely going BWOC and trying some composition(s) with $\frac{1}{z}$, and working away from $\lbrace 0 \rbrace$, gives you statements like $e^n \geq$ $| f(\frac{1}{n}) |$, or $e^n \geq |f(n)|$ which don't give you any sort of contradiction. So I think it will be some trick with bounding some expression coming from the power series, but I have no idea how to proceed.
Would really appreciate any help!