Let ( x n ) n ∈ N {\displaystyle {}{\left(x_{n}\right)}_{n\in \mathbb {N} }} and ( y n ) n ∈ N {\displaystyle {}{\left(y_{n}\right)}_{n\in \mathbb {N} }} be convergent real sequences with x n ≥ y n {\displaystyle {}x_{n}\geq y_{n}} for all n ∈ N {\displaystyle {}n\in \mathbb {N} } . Prove that lim n → ∞ x n ≥ lim n → ∞ y n {\displaystyle {}\lim _{n\rightarrow \infty }x_{n}\geq \lim _{n\rightarrow \infty }y_{n}} holds.
