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How can you prove the function $$f(x)=\sum_{k=1}^{\infty}\frac{(-1)^k}{k+|x|}$$ to be Lipschitz?

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We have \begin{align*} |f(x)-f(y)|&=\left|\sum_{k=1}^{+\infty}(-1)^k\left(\frac 1{k+|x|}-\frac 1{k+|y|}\right)\right|\\\ &=\left|\sum_{k=1}^{+\infty}(-1)^k\frac{|x|-|y|}{(k+|x|)(k+|y|)}\right|\\\ &\leq \sum_{k=1}^{+\infty}\frac{||x|-|y||}{(k+|x|)(k+|y|)}\\\ &\leq ||x|-|y||\cdot \sum_{k=1}^{+\infty}\frac 1{k^2}\\\ &\leq |x-y|\cdot \sum_{k=1}^{+\infty}\frac 1{k^2}. \end{align*}

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