Lemma 21.11.132F4F518BD56D

Proof. Write $F=\inf _{i\in I} F_ i$. We shall apply the characterization in 21.11.4. Take $n,r\in N$, $a,b\in \mathbb {R}$ such that $a>F(n)$, $b>F(r)$ and $t\in (0,1)$. We need to show that

By definition, there are $j_ n,j_ r\in I$ such that $a>F_ i(n)$ for $i\geq j_ n$ and $b>F_ i(r)$ for $i\geq j_ r$. Since $I$ is directed, we can find $j\in I$ dominating both $j_ n$ and $j_ r$, and then for any $i\geq j$, we have

It follows from 21.11.4 that

for any $i\geq j$. Since $F_ i$ is decreasing in $i$, we conclude (21.11.14). $\square$