Limit superior and limit inferior

I get the definition of limit superior and limit inferior, but I can't start a proof for these properties:

$\displaystyle \underline{\lim }{\,{x}_{n}}=-\overline{\lim }\left(-{{x}_{n}}\right)$

$\displaystyle \underline{\lim }{\,{x}_{n}}\le\overline{\lim }{\,{x}_{n}}$

$\displaystyle \underline{\lim }{\,{x}_{n}}+\underline{\lim }{\,{y}_{n}}\le\underline{\lim }\left({{x}_{n}}+{{y}_{n}}\right)\le\overline{\lim }{\,{x}_{n}}+\underline{\lim }{\,{y}_{n}}\le\overline{\lim }\left({{x}_{n}}+{{y}_{n}}\right)\le\overline{\lim }{\,{x}_{n}}+\overline{\lim }{\,{y}_{n}}.$

Hope you may enlight me. :D