Question: Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be a differentiable function with continuous uniformly derivative . Show:
$\lim\_{n \to \infty } n[f(x+1/n)-f(x)]=f'(x)$

I have tried the following: Since $f'(x)= \lim\_{h \to0 }[f(x+h)-f(x)/h]$ so if $h=1/n$ then $\lim\_{n \to \infty } n[f(x+1/n)-f(x)]=f'(x)$, I am well?

Answer: 
I used 'Elementary Probability', by D. Stirzaker, and 'A First Course on Probability', by S. Ross.
I heard from my friend who is a PhD student doing research in Probability, that he recommends "Real Analysis and Probability", by Dudley. Here is its  [amazon page](http://rads.stackoverflow.com/amzn/click/0521007542) .   
 He said it is clear, detailed, thorough, and doesn't skip steps. Also, it helped him transition from "undergraduate" probability to measure-theoretic probability. From his comments and the amazon reviews, it seems to match your criteria of "plenty of analysis", "focused on pure mathematics", and "plenty of rigorous proofs". I haven't read it yet, but it's definitely on my reading list.