Zipper theorem

The zipper theorem is a theorem about infinite convergent sequences.

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.

[edit] Proof

Assume that $a_n \to L$ and $b_n \to L$ . Let $c n$ be the new sequence $a 1$ , $b 1$ , $a 2$ , $b_2\dots$ $a n$ , $b_n\dots$

That is, let $c 2 n - 1$ = $a n$ for n ≥ 1 and $c 2 n = b n$ for $n \ge 1.$

Since $a_n\to L$ , for every $\varepsilon > 0$ there is an $N 1$ so that $|a_n-L| < \varepsilon$ when $n > N 1 .$

Since $b_n\to L$ , for every $\varepsilon > 0$ there is an $N 2$ so that $|b_n-L| < \varepsilon$ when $n > N 2 .$

If $N = max(N 1, N 2),$ then $|c_n-L|< \varepsilon$ whenever $n > 2 N$ and so $c n$ converges to $L .$

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This page was last modified 04:20, 17 January 2007 by Wikipedia user David Eppstein. Based on work by Wikipedia user(s) Oleg Alexandrov, Lunch, Gandalf61, JRSpriggs and Ghais and Anonymous user(s) of Wikipedia.
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