# Tails of Cauchy sequences, and their closures

Consider a metric space $$(S,d)$$, its completion $$\tilde S$$, and its closure $$\overline{S}$$. (By closure I imply: closure in completion.)

In the standard topology (one induced by the metric, i.e., based on open balls), we have $$\tilde{S} = \overline{S}$$. (Please correct me if I am wrong.)

Also consider a Cauchy sequence $$\left\{ y_n \right\}$$ and its tails $$Y_i\,=\,\left\{ y_i,\, y_{i+1},\;.\,.\,.\, \right\}\,\;.$$ The tails are nested, $$Y_{i+1} \subset Y_i\,\;,$$ but are not necessarily closed. Their diameters are defined as $$\operatorname{diam}\, Y_i\,\equiv\,\operatorname{sup}_{m,n\leq i}\,d(y_m,\,y_n)\,\;.$$ Likewise, the diameters of their completions (= closures) are $$\operatorname{diam}\, \overline{Y}_i\,\equiv\,\operatorname{sup}_{m,n\leq i}\,\left(\,d(y_m,\,y_n), \, d(y_m,\,y)\,\right)\,\;,$$ $$y$$ being the limit of $$\left\{y_i\right\}$$. This limit is not necessarily in the space $$S$$, but is in $$\tilde{S}$$.

The sequence is Cauchy, so the tails satisfy $$\operatorname{diam}\, {Y}_i\,\longrightarrow\,0\,\;.$$

QUESTION:

How to prove the same for the completions (= closures, on this occasion) -- i.e., to show that $$\operatorname{diam}\, {\overline{Y}}_i\,\longrightarrow\,0\,\;.$$

My understanding is that $$\operatorname{diam}\, \overline{Y}_i\,\leq \operatorname{diam}\, Y_i\,+\,\sup_{m\geq i}\,d(y_m,\,y)\,\;.$$ If this is correct, my question may be put in a simpler form: how to prove that $$\sup_{m\geq i}\,d(y_m,\,y)\,\longrightarrow\,0~~~\mbox{for}~~~i\longrightarrow\infty$$ Is this so simply by definition of the limit?

• I am unable to correct the sentence “In the standard topology, we have $\tilde{S} = \overline{S}$” since I don't know what it means. Feb 3 at 16:16
• What do you mean by the "closure" of $S$? $S$ is of course closed, i.e., $\operatorname{cl}_S S = S$. So you need to be more precise -- closure with respect to what?
– MPW
Feb 3 at 16:40
• Presumably, you mean the closure of $S$ in the completion? Feb 3 at 16:55
• @MPW Yes, good point, I shall explain that this a closure w.r.t. completion. Feb 3 at 17:26
• @ThomasAndrews Yes, in completion. I shall fix this in the text. Feb 3 at 17:27

HINT: What does it mean for $$y$$ to be the limit of the sequence $$(y_i)$$?