I have been a bit confused with how Cauchy sequences work. How might I construct a Cauchy Sequence to construct something non-obvious, like $\sqrt{2.5}$
I feel like there's something going over my head in regards to Cauchy sequences that I would like to understand.
In $\mathbb{R}$, you can define a sequence $a_{n}=\sqrt{2.5}+\frac{1}{n}$, to say that a sequence is Cauchy is to say that the the difference between successive terms becomes smaller and smaller. Assume WLOG $n>m\in\mathbb{N}$ /Then to prove that this sequence is Cauchy we need to show that given an $\epsilon$, there exists N such that $n,m > N$ implies $$|a_{n}-a_{m}|<\epsilon \Longrightarrow \left\lvert \left(\sqrt{2.5}+\frac{1}{n}\right)-\left(\sqrt{2.5}+\frac{1}{m}\right)\right\rvert <\epsilon$$ $$\left\lvert\frac{1}{n}-\frac{1}{m}\right\rvert< \epsilon \Longrightarrow \left\lvert\frac{m-n}{mn}\right\rvert=\left\lvert\frac{n-m}{mn}\right\rvert<\left\lvert\frac{n}{mn}\right\rvert=\left\lvert\frac{1}{m}\right\rvert<\epsilon\Longrightarrow N=\frac{1}{\epsilon}<m<n$$ . Therefore we have shown there exists an N such that the difference between successive terms can be made smaller than an arbitrary epsilon and the therefore the sequence is Cauchy.