Using lipschitz estimate to show $|f_n(x) - f_p(x) - (f_n(c)-f_p(c))| \leq |b-a|\sup_{y \in (a,b)}|f'_n(y)-f_p'(y)|$

Question

Assume $(f_n)$ is a sequence of functions that are continuous on $[a,b]$ and differentiable on $(a,b)$.

Then using Lipschitz estimate to prove that $$|f_n(x) - f_p(x) - (f_n(c)-f_p(c))| \leq |b-a|\sup_{y \in (a,b)}|f'_n(y)-f_p'(y)|$$

$\forall x \in [a,b]$ and all $n,p \in \mathbb{N}$

Furthermore, make it explicit which function you use lipschitz estimate on.

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The Lipschitz estimate is defined as such: Let $a < b$. Let $f$ be continuous on $[a, b]$ and differentiable on $(a, b)$. Then $|f(a) − f(b)| ≤ |b − a|\sup_{x∈(a,b)}|f'(x)|$

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How can I proceed with this?

I wanted to use the definition of continuity to make $|f_n(x) - f_n(c)|$ small, or to make $|f_p(x) - f_n(c)|$ small, but then I don't see how I can use lipschitz on the left-over terms.

Is anyone able to help me with this?

Thanks

Answer 1

Everything here is about $f_n-f_p$. So, first I would introduce $g=f_n-f_p$. Then the claim is $$|g(x) - g(c)| \leq |b-a|\sup_{y \in (a,b)}|g'(y)|$$ Since $x,c\in [a,b]$, this follows from the Mean Value Theorem.