TITLE: Prove that the curve of the function $ f $ has an asymptote near infinity which is parallel to the line $ y=x $
QUESTION [5 upvotes]: Let $ f $ be a continuous function from $ \Bbb R $ to $ \Bbb R $ satisfying the condition
$$(\forall x,y\in\Bbb R) \;|f(x)-f(y)|\ge |x-y|$$
If we assume that $ f $ is increasing and that
$$(\forall x\in \Bbb R)\;f(x)<x,$$
Prove that the curve $ C_f $ has an asymptot parallel to the line $ y= x, $ near $ +\infty$.
I tried to prove that there exists a real constant $ a$ such that
$$\lim_{x\to+\infty}(f(x)-x)=a $$
but i couldn't.
Any help will be appreciated.

REPLY [4 votes]: Let $g(x) = x - f(x)$. Then $g(x) > 0$. We show that $g(x)$ is monotone decreasing, which implies $\lim_{x \to \infty} g(x)$ exists, by the (continuous) monotone convergence theorem.
Suppose $y > x$. Then,
\begin{align*}
g(y) - g(x) &= (y - x) - (f(y) - f(x)) \\
&= (y - x) - |f(y) - f(x)| & \text{($f$ is increasing)} \\
&\le (y - x) - |y - x| \\
&\le 0,
\end{align*}
as required.