Prove that $\limsup _{n\to \infty}(a_n+b_n)\leq \limsup _{n\to \infty}a_n + \limsup _{n\to \infty}b_n$

Question

Let $(a_n)$ and $b_n$ be bounded sequences of real numbers. Prove that $$\limsup _{n\to \infty}(a_n+b_n)\leq \limsup _{n\to \infty}a_n + \limsup _{n\to \infty}b_n$$

How can this be proved?

Using the definition of limit, can I use the fact that $\sup|a_n + b_n|\leq \sup|a_n| + \sup|b_n|$?

Answer 1

Notice that $$ \limsup a_n= \lim_{n\rightarrow\infty}\sup_{m\geq n}a_m, $$ Thus, you can use the linearity of the limit together with $$ \sup_{m\geq n}(a_m+b_m)\leq \sup_{m\geq n}a_m+\sup_{m\geq n}b_m. $$ Right?