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

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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

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

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