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Suppose $A_n \to L$ and $B_n \to L$.

I need to show that the sequence $A_1,B_1,A_2,B_2,A_3,B_3,\dots$ converges to $L$. Now, I know that both $A_n$ and $B_n$ are less than or equal to $L$ for all $n$, however, how can I show that if you interleave the items in $A_n$ and $B_n$, it will also converge to $L$? It makes sense, I just don't know how to approach this problem.

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    Let $\epsilon>0$. Choose $N_1$ so that for $n> N_1$, $|A_n-L|<\epsilon$. Choose $N_2$ so that for $n> N_2$, $|B_n-L|<\epsilon$. What happens with the interleaved sequence if $n>2\max\{N_1,N_2\}$?2012-05-10
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    Choose $N = \max(N_A,N_B)$ so that it works for both the sequences.2012-05-10
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    You do **not** know that $A_n,B_n\le L$ for all $n$: the sequences might converge to $L$ from above, or they might bounce above and below $L$ while still converging to it.2012-05-10
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    @alexthebake: Is the fact that $A_n\leq L$ and $B_n\leq L$ an extra piece of information, or something that you are deducing? You can't deduce it, and it's also not necessary.2012-05-10

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