Prove that there exists $M>0$ such that for every $x\geq 1$ it satisfies $|f(x)|\leq Mx$

Question

Let $f:\mathbb R\rightarrow \mathbb R$ be a uniformly continuous function in the interval $[1,\infty)=I$.
Prove that there exists $M>0$ such that for every $x\geq 1$ it satisfies $|f(x)|\leq Mx$.
My attempt:

Let $\epsilon>0$. there exist $\delta>0$ such that for every $x,y\in I$, if $|x-y|<\delta$, then $|f(x)-f(y)|<\epsilon$(because f is uniformly continuous).

I know it's not much, but that's all I've got.