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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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I will suppose the sequence is monotone increasing (for the decreasing case just take negatives).

Suppose there is an $n\geq k$ such that $x_n \neq x_k$. Clearly $x_n > x_k$ by the monotonicity of $\left\{x_n\right\}$. Let $\epsilon = x_n - x_k$. By the convergence of $\left\{x_n\right\}$, there must be an $N$ such that whenever $j > N$, $x_j - x_k < \epsilon$. However, whenever $j > n$, $x_j - x_k \geq x_n - x_k = \epsilon$, so this $N$ cannot exist, contradicting the convergence of $\left\{x_n\right\}$. I conclude that $x_n = x_k$ for all $n \geq k$.

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