I tried using 2 mean value theorems but I got 2 different x values:
$(x-a)f_n'(c)$ and $(x-a)f_m'(d)$ so I couldn't make use of $f'_n$ unif convergence. What should I tweak?
2026-04-18 06:56:27.1776495387
${f_n}$ differentiable and $f'_n$ converges uniformly on $[a,b]$. How to show $f_n(x) - f_n(a)$ also converges?
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If $f_n'$ differs from $f_m'$ by less than $\epsilon$, then $f_n(x)-f_n(a)=\int_a^xf_n'(t)\,\mathrm dt$ disffers from $f_m(x)-f_m(a)=\int_a^xf_m'(t)\,\mathrm dt$ by less than $(x-a)\epsilon$.