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

\[\liminf _{n ightarrow \infty} x_{n} \leq \liminf _{n ightarrow \infty} y_{n}\]