In the above proof, it is taken that for both of the sequences $\exists N$ s.t $|a_n - L|<\epsilon $ and $|b_n - M|<\epsilon$. Is it correct to take the same $N$ for both sequences?
After that they have assumed the inequality which is true for $n\geq N_0$ is also true for $n\geq N$. Is that correct ?
They omitted some details, but it can be done. Let's say $|a_n - L| < \epsilon$ for $n \geq N_1$ and $|b_n - M| < \epsilon$ for $n \geq N_2$. Let $N = \max\{N_0,N_1,N_2\}$ and this $N$ would work for the whole proof (we just need the existence of such an $N$).