Interchange limit-integral & Equivalency of sequence convergence and concrete convergence?

Question

I got stuck in the proof of Cauchy's Integral Formula for higher derivatives in Stein's Complex Analysis, page 48:

Under what Conditions over a function $f$, we can infer that :

$\displaystyle\lim_{h\rightarrow 0}\int_Cf(z,h)dz=\int_C\lim_{h\rightarrow 0}f(z,h)dz$

A Try :

$\displaystyle\left|\int_Cf(z,h)dz-\int_C\lim_{h\rightarrow 0}f(z,h)dz\right|\leq P_C.\sup_{z\in C}\left|f(z,h)-\lim_{t\rightarrow 0}f(z,t)\right|$.

Now, I think there's a way to control right hand side, when $h$ would be small.

Answer 1

If you are unhappy with MCT and DCT, your proof is correct. The control you need is precisely what is called uniform convergence.

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With regards to your question in the comments,

Proof of $\lim_{h→ 0} f(h) = L$ iff for every null sequence $a_n$, $\lim_{n→∞} f(a_n) = L$.

Clearly LHS implies RHS. Suppose that it is not true that $\lim_{h→ 0} f(h) = L$. Then there is some $ε >0$ such that for every $\delta>0$, we can find a point $x<\delta$ such that $|f(x) - L| > ε $. Setting $\delta = 1/n$, we extract a null sequence $0<x_n<1/n$ of such points. Thus $f(x_n) \not→ L$.

This proof can be reused with almost no change for many 'limits', e.g. uniform limits of functions.