TITLE: A question on a Lipschitz function
QUESTION [5 upvotes]: This is the problem:
Prove or disprove the following statement:
If $f:[0,+\infty]\rightarrow\mathbb{R^+}$  is a Lipschitz function and not bounded, then it has necessarily $\lim_{x\to+\infty} f(x) = +\infty$

REPLY [3 votes]: Let
$$
T(x)=1-2\,|1-x|,\quad 0\le x\le1
$$
be the tent function. Now define $f$ as $f(x)=0$ if $0\le x\le1$ and
$$
f(x)=2^nT(2^{-n}(x-2^n))\quad \text{if}\quad 2^n\le x\le2^{n+1}.
$$

$f$ is Lipsthitz with constant $2$, unbounded, and $f(2^n)=0$ for all $n$, so that $f(x)\not\to\infty$. If you want $f$ to be strictly positive, just take $f(x)+1$.