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Suppose that $f$ is differentiable on $\mathbb{R}$. If $f(0)=1$ and $|f^{'}(x)|\leq1$ for all $x\in\mathbb{R}$, prove that $|f(x)|\leq|x|+1$ for all $x\in\mathbb{R}$.

I tried:

Let $g(x)=|f(x)|-|x|-1$. Then I tried to find $g^{'}(x)$ but I'm not sure where to start.

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