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**dwsmith** Suppose $\displaystyle \{x_n\}\to x_0$ and $\displaystyle \{y_n\}\to x_0$. Define a sequence $\displaystyle \{z_n\}$ as follows: $\displaystyle z_{2n}=x_n$ and $\displaystyle z_{2n-1}=y_n$. Prove that $\displaystyle \{z_n\}$ converges to $\displaystyle x_0$.

Let $\displaystyle \epsilon >0$. Then $\displaystyle \exists N_1, \ N_2\in\mathbb{N}$ such that for $\displaystyle n\geq N_1, \ N_2$ we have $\displaystyle |x_n-x_0|<\epsilon$ and $\displaystyle |y_n-x_0|<\epsilon$.