Exercise 2.3.5

Question

Let $\left(x_{n}\right)$ and $\left(y_{n}\right)$ be given, and define $\left(z_{n}\right)$ to be the ``shuffled'' sequence $\left(x_{1}, y_{1}, x_{2}, y_{2}, x_{3}, y_{3}, \ldots, x_{n}, y_{n}, \ldots\right)$. Prove that $\left(z_{n}\right)$ is convergent if and only if $\left(x_{n}\right)$ and $\left(y_{n}\right)$ are both convergent with $\lim x_{n}=\lim y_{n}$.

Ulisse Mini · Accepted Answer

Obviously if $\lim x_n = \lim y_n = l$ then $z_n 	o l$. To show the other way suppose $(z_n) 	o l$, then $|z_n - l| < \epsilon$ for all $n \ge N$ meaning $|y_n - l| < \epsilon$ and $|x_n - l| < \epsilon$ for $n \ge N$ aswell. Thus $\lim x_n = \lim y_n = l$.

Exercise 2.3.5

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Exercise 2.3.5

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