If $f$ is monotonic and its limits at $\pm\infty$ are equal, then $f$ is constant?

Question

I need to prove or disprove this. It looks correct but I can't prove it.

If $f(x)$ is monotonic and $\lim _{x\to \infty } f(x)=\lim _{x\to -\infty }f(x)=L$ then $f$ is constant.

Answer 1

Argue by contradiction: say there are values $x<y$ such that WLOG $f(x)<f(y)$ and use the definition of limit at $\infty$ or $-\infty$ to conclude that there must be either a $y<z$ with $f(y)>f(z)$ or a $z<x$ with $f(z)>f(x)$, in either case contradicting monotonicity.

Answer 2

Without loss of generality, suppose $f$ is non-decreasing.

Suppose by contradiction $f$ is not constant, i.e. that there exist $x<y$ such that $f(x) < f(y)$. Then show that $L\leq f(x)$ and $f(y)\leq L$: Since $$\forall t \leq x, f(x) \geq f(t) \tag{monotonicity} $$ taking limits as $t\to\infty$ we get $f(x) \geq \lim_{t\to-\infty} f(t) = L$. (Similar for the other one.)

So $L < L$, contradiction.