EDB — 02M

[02M]Difficulty:*. ¹ Consider the following quotient of the family of all integer valued sequences

where we define \(a∼ b\) iff \(a_ k= b_ k\) eventually in \(k\).

We define the ordering

that is, \(a⪯ b\) when \(a_ k≤ b_ k\) eventually. This is a preorder and

so it passes to the quotient were it becomes an ordering, see Prop. [1Z7]↺↻.

Let \(a^ k\) be an increasing sequence of sequences, that is, \(a^ k⪯ a^{k+1}\); we readily see that it has an upper bound \(b\), by defining

We can then apply the Zorn Lemma to assert that in the ordered set \(({\mathbb X},⪯)\) there exist maximal elements.

Given \(a,b\) we define

then it is easily verified that \(a⪯ a∨ b\). So this a direct ordering, see [06N]↺↻.

We conclude that the ordered set \(({\mathbb X},⪯)\) has an unique maximum, by [06S]↺↻.

This is though false, since for any sequence \(a\) the sequence \((a_ n+1)_ n\) is larger than that.

What is the mistake in the above reasoning? What do you conclude about \(({\mathbb X},⪯)\)?