Zipper theorem

The zipper theorem is a theorem about infinite convergent sequences. :If $\{a\_n\}$ and $\{b\_n\; \}$ both converge to $L$, then the sequence $a\_1$, $b\_1$ , $a\_2$ , $b\_2$ , . . . , $a\_n$ , $b\_n$ , . . . converges to L.

Proof

Assume that $a\_n\; o\; L$ and $b\_n\; o\; L$. Let $c\_n$ be the new sequence $a\_1$, $b\_1$ , $a\_2$ , $b\_2dots$ $a\_n$ , $b\_ndots$

That is, let $c\_\{2n-1\}$ = $a\_n$ for n ≥ 1 and $c\_\{2n\}\; =\; b\_n$ for $n\; ge\; 1.$

Since $a\_n\; o\; L$, for every $varepsilon\; >\; 0$ there is an $N\_1$ so that when $n\; >\; N\_1.$

Since $b\_n\; o\; L$, for every $varepsilon\; >\; 0$ there is an $N\_2$ so that when $n\; >\; N\_2.$

If $N\; =\; max\; (N\_1,\; N\_2),$ then whenever $n\; >\; 2N$ and so $c\_n$ converges to $L.$