I'm trying to do the following exercise for my Real Analysis class:
Let $p$ be a given natural number. Give an example of a sequence $\left(x_{n}\right)$ that is not a Cauchy sequence, but that satisfies $\lim \left|x_{n+p}-x_{n}\right|=0$
However, I am in constant doubt in regard to the "let" word in math texts. Can I choose, say $p = 1$, or when one says "let" I am supposed to stick with $p \in \mathbb{N}$ and nothing more? What kind of "control" do I have over $p$?
For instance, when one says "Given $\epsilon \gt 0$", I usually see things like: Let $\epsilon = \frac{\epsilon}{2}$ so one can finish a certain argument.
Can someone help me? I need to solve this doubt once and for all, it bothers me very often.
from Hot Weekly Questions - Mathematics Stack Exchange
Lucas
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