Show that the sequence converges to L

• Mar 22nd 2010, 05:29 PM
wopashui
Show that the sequence converges to L
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.
• Mar 23rd 2010, 04:18 AM
tonio
Quote:

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