If $f \colon [a,\infty) \longrightarrow\mathbb R$, $f$ monotonically increasing and $\lim_{x\to \infty} f(x)$ exists then it is also upper bounded

Question

Prove that if $f\colon[a,\infty)\longrightarrow R$, $f$ monotonically increasing and $\lim_{x\to \infty} f(x)$ exists then it is also upper bounded

How do I prove this?

The only thing that I could find was this but it requires to prove that limit exists given that f is already bounded above and increasing.

Answer 1

Let $M=\lim_{x\to+\infty}f(x)$. Could there be a $x_0\in[a,+\infty)$ such that $f(x_0)>M$? No, because $f$ is monotonically increasing and\begin{align}f(x_0)>M&\implies\bigl(\forall x\in[x_0,+\infty)\bigr):f(x)\geqslant f(x_0)>M\\&\implies\lim_{x\to+\infty}f(x)\geqslant f(x_0)>M,\end{align}which is a contradiction, since $M=\lim_{x\to+\infty}f(x)$.

Answer 2

Hint: Try a proof by contradiction. If $f$ were not bounded from above and monotonically increasing, can you show that $\lim_{x\to\infty} f(x)$ does not exist?