Let $f$ and $g$ be two functions from $[0,1]$ to $\mathbb R$. They are continuous on $[0,1]$ and differentiable on $(0,1)$. If $\|f'(t)\| \le g'(t)$ at all points of the interval, prove that $|f(1) - f(0)| \le g(1) - g(0)$. I used the mean-value theorem to get that $|f(1) - f(0)| \le |f'(c)|$ for some number $c$ in the interval $(0,1)$ and i applied the same theorem on $g$ to get $|g(1) - g(0)| \le |g'(c)|$. I need help in combining the results to prove that $|f(1) - f(0)| \le g(1) - g(0)$.

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    $\begingroup$ There's a different thing to use here: Grönwall's lemma. $\endgroup$ – AlexR Mar 4 '15 at 17:36

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