Prove uniformly continuity at $\infty$ to continuous function

Question

Say $f:[0,\infty) \to \mathbb{R}$ is a continuous function. Assume $\lim_{x \to \infty}[f(x)-ax]=b$ for some $a,b \in \mathbb{R}$ and prove $f$ is uniformly continuous in $[0, \infty)$

So if the set was closed than Cantor will do the trick. I've tried proving $\lim_{x \to \infty}f(x)$ exists, or using the definition of uniform continuity and using the continuity of $f(x)$ but to no ends. Any hints?

Answer 1

The function $g(x) = f(x) - ax - b$ is continuous on $[0,\infty)$ such that $\lim_{x\to \infty} g(x) = 0$, so $g$ is uniformly continuous. Since the function $h(x) = ax + b$ is uniformly continuous on $[0,\infty)$, then $f$, being the sum of $g$ and $h$, must be uniformly continuous.

Answer 2

For any $\epsilon>0$ there exists some $K>0$ such that $|f(x)-ax-b|<\epsilon$ for $x>K$.

Now show the uniform continuity on $[0,K]$ and on $(K,\infty)$.