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.
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