# Thread: Show that the sequence converges to L

1. ## 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.

2. 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...

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