Let \(\left\{x_{n}ight\}_{n=1}^{\infty}\) and \(\left\{y_{n}ight\}_{n=1}^{\infty}\) be a sequences of real numbers such that

\[\left|\limsup _{n ightarrow \infty} x_{n}ight|<\infty\]

and

\[\left|\limsup _{n ightarrow \infty} y_{n}ight|<\infty\]

Then prove that

\[\liminf _{n ightarrow \infty} x_{n}+\liminf _{n ightarrow \infty} y_{n} \leq \liminf _{n ightarrow \infty}\left(x_{n}+y_{n}ight)\]

and

\[\limsup _{n ightarrow \infty}\left(x_{n}+y_{n}ight) \leq \limsup _{n ightarrow \infty} x_{n}+\limsup _{n ightarrow \infty} y_{n} .\]