If $\limsup\limits_{n\to\infty}a_n=+\infty$ and $\limsup\limits_{n\to\infty}b_n\in\mathbb{R}$, then $\limsup\limits_{n\to\infty}(a_n+b_n)=+\infty$? There is the following exercise in a book:

$\limsup \limits_{n \to \infty} (a_n + b_n) \leq \limsup \limits_{n \to \infty} a_n + \limsup \limits_{n \to \infty} b_n $

And the author(Kazuo Matsuzaka) writes as follows in his solution without proofs:

If $\limsup \limits_{n \to \infty} a_n = +\infty$ and $\limsup \limits_{n \to \infty} b_n = +\infty$, then $\limsup \limits_{n \to \infty} (a_n + b_n) = +\infty$.
If $\limsup \limits_{n \to \infty} a_n = +\infty$ and $\limsup \limits_{n \to \infty} b_n \in \mathbb{R}$, then $\limsup \limits_{n \to \infty} (a_n + b_n) = +\infty$.

Please tell me how to prove the above two statements if they are true.
Please tell me counter examples if the above two statements are false.

The first is easy to prove:
Since,for all $m \in \mathbb{N}$:
$$
\limsup_{n\to\infty}a_n \leq \sup_{k\geq m} a_k 
$$
If:
$$
\limsup_{n\to\infty}a_n = +\infty
$$
Then:
$$
\sup_{k\geq m} a_k = +\infty
$$
Therefore if:
$$
\limsup_{n\to\infty}a_n = +\infty\\
\limsup_{n\to\infty}b_n = +\infty
$$
Then, for all $k \in \mathbb{N}$:
$$
\sup_{k\geq m}a_n = +\infty\\
\sup_{k\geq my}b_n = +\infty
$$
Therefore:
$$
\sup_{k\geq m}a_n > 0\\
\sup_{k\geq m}b_n > 0
$$
Therefore:
$$
\sup_{k\geq m}a_n + \sup_{k\geq m}b_n > \sup_{k\geq m}a_n
$$
By the linearity of $\sup$:
$$
\sup_{k\geq m}(a_n + b_n) > \sup_{k\geq m}a_n
$$
Taking limits on $k$(allowable since the inequality is valid for all $k \in \mathbb{N}$):
$$
\lim_{k\to\infty}\sup_{k\geq m}(a_n + b_n) > \lim_{k\to\infty}\sup_{k\geq m}a_n
$$
Thus, by the definition of $\limsup$:
$$
\limsup_{n\to\infty}(a_n + b_n)  > \limsup_{n\to\infty}a_n
$$
Since:
$$
\limsup_{n\to\infty}a_n = +\infty
$$
Therefore:
$$
\limsup_{n\to\infty}(a_n + b_n) = +\infty
$$
I'll have to think about the second statement.
