We are given that $(a_n)$ converges to $L$ and $(b_n)$ converges to $M$. By definition of convergence, for any $\epsilon > 0$, there exist positive integers $N_1$ and $N_2$ such that for all $n \geq N_1$, $|a_n - L| < \frac{\epsilon}{2}$ and for all $n \geq N_2$,
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