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elementary analysis proof verification

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Show that if X, Y and Z are convergent sequences, then the sequence $w:=mid\{x_n,y_n,z_n\}$ is convergent.

The attempt:

(1) Suppose $x_n$ goes to x, $y_n$ goes to y, and $z_n$ goes to z. WLOG, assume $x \leq y \leq z$.

(2) There exists an $N$ such that $n \geq N$ implies that $x_n \leq y_n \leq z_n$.

(3) Therefore, for $n \geq N$, we have $w:=mid\{x_n,y_n,z_n\}=y_n$, which converges.

(4) Since the tail end of $w_n$ converges, the sequence $w_n$ also converges.

I know, the 2nd step requires extra work, i tried and couldn't show it. However, my intuition tells me it is true...


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