At the end of class on Friday, Jeff and I had an interesting discussion about the wording of Theorem 1.60, which says

If $p_1, p_2, p_3, \ldots$ is a non-decreasing sequence and there is a point, $x$, to the right of

eachpoint of the sequence, then the sequence converges to some point.

Jeff claimed (please correct me if I am wrong) that the word **every** should have been used (as it is in the Completeness Axiom) instead of **each**. I claimed that it didn't matter which word was used and that the meaning was the same. I've just thought about this a little more and figured that I might as well share my thoughts.

The quantifier on $x$ indicates that we are talking about a single $x$. This single $x$ is to right of **each** point in the sequence. That is, if you pick a random $p_i$ in the sequence, $p_i<x$. This implies that $x$ is to the right of every point in the sequence. The $x$ is a "one size fits all" point. So, I maintain it doesn't matter whether we use **each** or **every**.

Contrast the statement of the theorem with the following statement:

For each point in the sequence, there is a point $x$ that is to the right.

In this case, the dependence of the quantifiers is reversed and we no longer have a "one size fits all" $x$. Each individual point $p_i$ in the sequence has a point $x_i$ to the right, where I am using a subscript on $x$ to emphasize that this point depends on $p_i$. The point to the right of $p_i$ may be different than the point to the right of $p_j$ (assuming $i\neq j$). This is a very different situation than that in Theorem 1.60.