On page 37 of "The Foundations of Mathematics" (2nd edition) the authors write of the reals...
We can make the completeness property of the reals very plausible in terms of our ideas about decimals. Let $(a_n)$ be an increasing sequence of real numbers, with $a_n \leq k$ for all $k$.
(My first question: doesn't this condition mean $a_n$ is less than any number? I am already confused, but maybe it should say "all $n$"?)
Continuing...
The set of integers between $a_n - 1$ and $k$ is finite, so there is an integer $b_0$ that is the largest integer for which some term $a_{n_0}$ of the sequence is $\geq b_0$. Now all terms $a_n$ are less than $b_0 + 1$.
I'm afraid I really don't follow this at all, surely there are (potentially) a very large number of such integers? I suspect I have not grasped what the authors are trying to say at all. Could someone explain, thanks?
It should be $n$ instead of $k$.
If for some $m$, $a_m \ge b_0+1$, then $b_0+1$ is less than or equal to the largest integer $s$ such that for some term (here $m$) of the sequence, $a_m \ge s$. But this integer $s$ is $b_0$ by assumption. Contradiction