real analysis - Limit of convergent monotone sequence

Looking for a nice proof for this proposition:

Let $\{ x_n \}$ be a convergent monotone sequence. Suppose there
exists some $k$ such that $\lim_{n\to\infty} x_n = x_k$, show that
$x_n = x_k$ for all $n \geq k$.

I have the intuition for why it's true but am having a tough time giving a rigorous proof.