/sci/ - Science & Math

>>11326270
Ah, I see.
We call the set [math]S \subset \mathbb{R}[/math]. Since [math]S[/math] is nonempty, [math]a \in S[/math]. [math]S[/math] also has a lower bound [math]b \in \mathbb{R}[/math].
We consider [math]a', ~ b'[/math] to be the integer parts of [math]a[/math] and [math]b[/math].
There are a finite number of integers between [math]a'[/math] and [math]b'[/math]. We pick the largest one which is also a lower bound, [math]c[/math].
Now, we pick the largest [math]n \in \mathbb{N}[/math] such that [math]c.n[/math] is also a lower bound, where n can be zero.
By iterating this procedure infinitely for the rest of the decimal cases, we produce a number [math]i[/math], which is the infima.
First, if [math]d \in S[/math] and [math]d<i[/math], then it is larger in some decimal case. This can't happen because of the construction. Thus, [math]i[/math] is a lower bound.
If some other [math]e[/math] is larger than [math]i[/math], then it is larger in some decimal case. Thus, by construction, it isn't a lower bound.