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.