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

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 $\sup|a_n + b_n|\leq \sup|a_n| + \sup|b_n|$?

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This question has already been asked, I think? Check this out. –  Lays Oct 4 '13 at 22:24
Don't use absolute values, the end result will be wrong if you do that (if the sequence has negative numbers for instance) –  Evan Oct 4 '13 at 23:04
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?