Understanding the Use of Min in Cauchy Sequences

[yucheng]

Homework Statement

Proposition 5.3.10 (Multiplication is well defined). Let $x = \mathrm{LIM}_{n\to\infty} a_n$, $y = \mathrm{LIM}_{n\to\infty} b_n$, and $x' = \mathrm{LIM}_{n\to\infty} a'_n$ be real numbers. Then xy is also a real number. Furthermore, if x=x', then xy=x'y.

Relevant Equations

N/A

I refer to this page: https://taoanalysis.wordpress.com/2020/03/26/exercise-5-3-2/

Since $(a_n)_{n=1}^\infty$ is Cauchy, for $\varepsilon' := \min\left(\frac{\varepsilon}{3M_2}, \frac\varepsilon 3\right)$ we see that this sequence is eventually $\varepsilon'-steady$. Similarly, since $(b_n)_{n=1}^\infty is Cauchy$, for $\delta := \min\left(\frac{\varepsilon}{3M_1}, 1\right)$ the sequence is eventually $\delta$-steady.

I am having trouble understanding the purpose / motivation behind using the min as in $\delta := \min\left(\frac{\varepsilon}{3M_1}, 1\right)$ and $\varepsilon' := \min\left(\frac{\varepsilon}{3M_2}, \frac\varepsilon 3\right)$. Please enlighten me.

Thanks in advance.

Edit:
As per @Stephen Tashi 's suggestion, Tao defines a real number, $x$, $x=\operatorname{LIM_{n\to\infty}} a_n$ which is the formal limit of a Cauchy sequence $(a_n)_{n=1}^\infty$. A Cauchy sequence being for each $\epsilon>0$, we can find an N such that for all $j,k\geq N$, we have $|a_j-a_k|\leq \epsilon$.

P.S. formal limit is very much akin to limit. It is just a "scaffold" as Tao puts it, that will be replaced by the notion of limits.

 

[Stephen Tashi]

I suggest you explain the proposition to be proved by giving the necessary background information. In particular, what is the definition of the "LIM" concept?

In glancing at a video review of Terrance Tao's analysis books, I find that at 3:03 in
we see a page that says

However, unlike our work in constructing the integers (where we eventually replaced formal differences with actual differences) and rationals (where we eventually replaced formal quotients with actual quotients) we never really finished the job of constructing the real numbers, because we never got around to replacing formal limits $LIM_{n \rightarrow \infty} a_n$ with actual limits $lim_{n_\rightarrow \infty} a_n$.