Limit assumed to exist practice

$\limsup(a_{n} + b_{n}) \le \limsup a_{n} + \limsup b_{n}$

$\limsup(a_{n} + b_{n}) = \limsup a_{n} + \limsup b_{n}$

If $b_{n}$ converges then $\limsup(a_{n} + b_{n}) = \limsup a_{n} + \lim b_{n}$

$\liminf a_{n} = -\limsup(-a_{n})$