# Real numbers/Convergent sequences/Compare/Fact/Proof/Exercise

Let and be convergent real sequences with for all . Prove that holds.

Let ${}{\left(x_{n}\right)}_{n\in \mathbb {N} }$ and ${}{\left(y_{n}\right)}_{n\in \mathbb {N} }$ be convergent real sequences with ${}x_{n}\geq y_{n}$ for all ${}n\in \mathbb {N}$. Prove that ${}\lim _{n\rightarrow \infty }x_{n}\geq \lim _{n\rightarrow \infty }y_{n}$ holds.