Suppose that $\displaystyle a_n --> L $and $\displaystyle b_n --> L$. Show that the sequence
$\displaystyle a_1,b_1,a_2,b_2,a_3,b_3$,...
converges to L.
Let $\displaystyle \epsilon > 0\Longrightarrow \exists\,N_1,\,N_2\in\mathbb{N}\,\,\,s.t.\,\,\,|a_ n-L|<\epsilon\,\,\forall\,n>N_1\,\,\,and\,\,\,|b_n-L|<\epsilon\,\,\forall\,n>N_2$ .
Let $\displaystyle M:=\max(N_1,N_2)$ , then $\displaystyle \forall\,n>M\,,\,\,|x_n-L|<\epsilon$ , with $\displaystyle x_n=a_n\,\,\,or\,\,\,b_n$ (either one, it doesn't matter), so...
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