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**ymar** Let $\displaystyle f:[0,\infty]\longrightarrow \mathbb{R}$ be a continuous function and $\displaystyle \lim_{x\rightarrow\infty}(f(x)-(ax+b))=0$. Show that $\displaystyle f$ is uniformly continuous.

I'd like to use this charactetization of uniform continuity. A function $\displaystyle f$ is uniformly continuous iff for all sequences $\displaystyle x_n,\,y_n$ if $\displaystyle |x_n-y_n|\rightarrow 0$, then also $\displaystyle |f(x_n)-f(y_n)|\rightarrow 0$.

Suppose $\displaystyle |x_n-y_n|\rightarrow 0$.

If $\displaystyle x_n$ is a bounded sequence, then $\displaystyle y_n$ must be bounded too, and so if $\displaystyle |x_n-y_n|\rightarrow 0$, then also $\displaystyle |f(x_n)-f(y_n)|\rightarrow 0$, because $\displaystyle f$ is continuous and therefore uniformly continuous on a compact set. If, on the other hand, $\displaystyle x_n\rightarrow\infty$, then also $\displaystyle y_n\rightarrow\infty$ and we have

$\displaystyle |f(x_n)-f(y_n)|\leq |f(x_n)-(ax_n+b)|+|f(y_n)-(ay_n+b)|+|(ax_n+b)-(ay_n+b)|\rightarrow 0$

But what if $\displaystyle x_n$ is neither bounded nor convergent to infinity?