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- Apr 13, 2013

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I have a question..I am given the following exercise:

Let $f:[0,+\infty) \to \mathbb{R}$ uniformly continuous at $[0,+\infty)$.Prove that there are $a,b \geq 0$ such that $|f(x)| \leq ax+b , \forall x \geq 0$.

That's what I did so far:

$f:[0,+\infty) \to \mathbb{R}$ is uniformly continuous at $[0,+\infty)$.

So: $\forall \epsilon>0 \exists \delta>0 $ such that $\forall x,y \in [0,+\infty)$ with $|x-y|<\delta \Rightarrow |f(x)-f(y)|<\epsilon$..

But,how can I continue to show that there are $a,b \geq 0$ such that $|f(x)| \leq ax+b , \forall x \geq 0$??