How can I prove that if $f:[0,+\infty)\to[0,1]$ is continuous and increasing, then it is uniformly continuous? [closed]

The problem is the following:Let $f:[0,+\infty)\rightarrow\mathbb{R}$ such that it is continuous, strictly growing and its image is contained in $[0,1]$, is uniformly continuous.

I've tried in several ways but none of them convinces me because I can never get to use all hypothesis. I want to know how can I write the solution to this problem in an understandable (and correct) way.