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Let $f:\mathbb R\to\mathbb R$ be continuous. Suppose $(x_n)_n$ and $(y_n)_n$ are sequences in $\mathbb R$ such that the sequence $(x_n-y_n)_n$ converges to $0$. Does this mean that the sequence $(f(x_n)-f(y_n))_n$ converges to $0$?

I feel like it is true, since the definition of continuity states that $f$ preserves limits of convergent sequences, but I do not know how to prove it.

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    Try thinking about continuous functions that are not uniformly continuous. Based on your remark, to find a counterexample you should make sure that $(x_n)$ and $(y_n)$ are not convergent sequences.2012-05-22
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    @Jonas Meyer thank you for your help, I've just found a counterexample.2012-05-22

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