Let \(\left\{x_{n}ight\}_{n=1}^{\infty}\) and \(\left\{y_{n}ight\}_{n=1}^{\infty}\) be a sequences of real numbers such that \(x_{n}>0\) and \(y_{n}>0\) for all \(n \in \mathbb{N}\),

\[0<\limsup _{n ightarrow \infty} x_{n}<\infty\]

and

\[0<\limsup _{n ightarrow \infty} y_{n}<\infty\]

Prove that

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