# Real sequences/Alternation/Convergence/Exercise

Let and be sequences of real numbers and let the sequence be defined as and . Prove that converges if and only if and converge to the same limit.

Let ${}(x_{n})_{n\in \mathbb {N} }$ and ${}(y_{n})_{n\in \mathbb {N} }$ be sequences of real numbers and let the sequence ${}(z_{n})_{n\in \mathbb {N} }$ be defined as ${}z_{2n-1}:=x_{n}$ and ${}z_{2n}:=y_{n}$. Prove that ${}(z_{n})_{n\in \mathbb {N} }$ converges if and only if ${}(x_{n})_{n\in \mathbb {N} }$ and ${}(y_{n})_{n\in \mathbb {N} }$ converge to the same limit.