I am not very familiar with mathematical proofs, or the notation involved, so if it is possible to explain in 8th grade English (or thereabouts), I would really appreciate it.
Since I may even be using incorrect terminology, I'll try to explain what the terms I'm using mean in my mind. Please correct my terminology if it is incorrect so that I can speak coherently about this answer, if you would.
Sequential infinite set: A group of ordered items that flow in a straight line, of which there are infinitely many. So, all integers from least to greatest would be an example, because they are ordered from least to greatest in a sequential line, but an infinite set of bananas would not, since they are not linearly, sequentially ordered. An infinite set of bananas that were to be eaten one-by-one would be, though, because they are iterated through (eaten) one-by-one (in linear sequence).
Sequential infinite subsets: Multiple sets within a sequential infinite set that naturally fall into the same order as the items of the sequential infinite set of which they are subsets. So, for example, the infinite set of all integers from least to greatest can be said to have the following two sequential infinite subsets within it: all negative integers; and all positive integers. They are sequential because the negative set comes before the positive set when ordered as stated. They are infinite because they both contain an infinite qty of items, and they are subsets because they are within the greater infinite set of all integers.
So I'm wondering if every (not some, but every) sequential infinite set contains within it sequential infinite subsets. The subsets (not the items within them) being sequentially ordered is extremely important. Clearly, a person could take any infinite set, remove one item, and have an infinite subset. Put the item back, remove a different item, and you have multiple infinite subsets. But I need them to be not only non-overlapping, but also sequential in order.
Please let me know if this does not make sense, and thank you for dumbing the answer down for me.