Maybe we should talk about what "infinite" means. I'd like to propose the following idea: a sequence of things is infinite, if there is always "one more" object to consider. We could also say that for any number of finite steps, there is always another object of the series we haven't looked at yet.

As an example, the sequence of natural numbers would satisfy this: if I start considering the sequence 1, 2, 3 and so on, if I ever stop after finite time (say 1729 steps), I can always compute +1 to find another element of the sequence I haven't seen yet.

Also consider the following: the set of all numbers between 0 and 1 is in some sense bounded. However, I can find an infinite sequence of numbers in this set: consider 1/1, 1/2, 1/3, 1/4, ...

These numbers are always between 0 and 1, and are infinitely decreasing.

Perhaps the confusion comes from you talking about infinity as in a number which is larger than any real or natural number, while I'm talking about sizes of sets of infinite size. As I had demonstrated earlier, we can show the existence of uncountable infinite vs countably infinite sets, while such distinctions don't really come up in limit theory and calculus.