If $f: [1, \infty ) \rightarrow \mathbb{R}$ is uniformly continuous, prove that $f(x)$ is bounded by some positive, finite $M$ times $x$
$f$ defined on $[1,\infty )$ is uniformly continuous. Then $\exists M>0$ s.t. $\frac{|f(x)|}{x}\le M$ for $x\ge 1$.