# Cantor diagonal process to extract a subsequence of continuously differentiable functions

Let $$\left(f_{h}\right)_{h} \subset \mathcal{C}^{1}([a, b]) .$$ Suppose we have by assumption the sets of functions $$\left\{f_{h}: h \in \mathbb{N}\right\} \quad and \quad\left\{f_{h}^{\prime}: h \in \mathbb{N}\right\}$$ relatively compact in $$\left(\mathcal{C}^{0}([a, b]),\|\cdot\|_{\infty}\right)$$ Then $$\exists\left(h_k\right)_{k}$$ increasing sequence of integers and $$\exists g_{1}, g_{2} \in \mathcal{C}^{0}([a, b])$$ such that $$f_{h_{k}} \longrightarrow g_{1} \text { for } k \rightarrow+\infty \text { in }\left(\mathcal{C}^{0}([a, b]),\|\cdot\|_{\infty}\right)$$ and $$f_{h_{k}}^{\prime} \longrightarrow g_{2} \text { for } k \rightarrow+\infty \text { in }\left(\mathcal{C}^{0}([a, b]),\|\cdot\|_{\infty}\right)$$

What I have is that the $$f_{h_{k}}^{\prime}$$ now are actually not the derivatives of the $$f_{h_{k}}$$ and my goal is to be able to extract from the sequences $$f_{h_{k}}$$ a subsequence such that the $$f_{h_{k}}^{\prime}$$ of my new subsequence are actually their derivatives. My goal is to apply the Cantor diagonal procedure on a dense set of $$[a,b]$$, but I have difficulties in formalizing it. Thanks in advance!

No need for diagonal procedure. First pick a subsequence along which $$(f_h)$$ converges in the norm. Look at the derivatives along this subsequence. There is a further subsequence along which $$(f_h')$$ converges. You now have one subsequence along which both $$(f_h)$$ and $$(f_h')$$ converge.
Now use the fact that $$f_{h_k} (x)=f_{h_k} (c)+\int_c^{x} f_{h_k}' (y)dy$$ to conclude that $$g_1$$ is differentiable and its derivative is $$g_2$$.