If the limit of the sum exists, then the limit of each component exists

Question

Consider two sequences $\{b_n\}_{n\in \mathbb{N}}$ and $\{a_n\}_{n\in \mathbb{N}}$. Suppose that $$ (*) \hspace{1cm} \lim_{n\rightarrow \infty} (b_n+a_n)=L<\infty $$

Does this imply $$ \exists \lim_{n\rightarrow \infty} b_n \text{ and it is finite} $$ $$ \exists \lim_{n\rightarrow \infty} a_n \text{ and it is finite} $$ so that (*) is equivalent to $$ \lim_{n\rightarrow \infty}b_n=-\lim_{n\rightarrow i\infty} a_n+L $$ ?

Answer 1

No. E.g let $a_n=n$ and $b_n=-n$.

Answer 2

It can't be true in general, but it is always true if the two sequecences are monotonic and with constant sign.