I'm trying to prove a theorem in this page : http://en.m.wikipedia.org/wiki/Differentiation_under_the_integral_sign#/search
Everything is completely fine except one line.
How do i prove that "For all $\epsilon>0, (\exists \Delta \alpha)\forall x\in[a,b], |f(x,\alpha+\Delta \alpha)-f(x,\alpha)|<\epsilon$?
It's written there that it's a consequence of Heine-Cantor theorem. But how?
Let's fix $\alpha$ and $\epsilon$.
Then, there exists $\delta>0$ such that $d(x,y)<\delta \Rightarrow |f(x,\alpha)-f(y,\alpha)|<\epsilon$
This is strictly weaker than the statement written above. How could i prove that??