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.