Show that the sequence converges to L

**wopashui** · March 22nd 2010, 05:29 PM

Suppose that $a_n --> L$ and $b_n --> L$ . Show that the sequence

$a_1,b_1,a_2,b_2,a_3,b_3$ ,...

converges to L.

**tonio** · March 23rd 2010, 04:18 AM

Originally Posted by wopashui

Suppose that $a_n --> L$ and $b_n --> L$ . Show that the sequence

$a_1,b_1,a_2,b_2,a_3,b_3$ ,...

converges to L.

Let $\epsilon > 0\Longrightarrow \exists\,N_1,\,N_2\in\mathbb{N}\,\,\,s.t.\,\,\,|a_ n-L|<\epsilon\,\,\forall\,n>N_1\,\,\,and\,\,\,|b_n-L|<\epsilon\,\,\forall\,n>N_2$ .

Let $M:=\max(N_1,N_2)$ , then $\forall\,n>M\,,\,\,|x_n-L|<\epsilon$ , with $x_n=a_n\,\,\,or\,\,\,b_n$ (either one, it doesn't matter), so...

Tonio

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