Question

If you can not reach infinite, and there are infinite numbers between any two integers, how do we know the next integer exists?

Answer 1

we don't
is a conceptual modality

Answer 2

just like infinity, is also a concept
and is prone to any other logical inconsistencies

Answer 3

Use a summation. Say we want to prove that the integer 1 exists.
We can define a summation:
.9+.09+.009+...
as
\[\sum_{n=0}^{\infty}.9(.1^n)\]
This is a geometric series, which is exactly equal to a/(1-r), where a is the first value(.9) and r is the ratio(.1)
.9/(1-.1) = .9/.9 = 1 exactly. If you take the summation of numbers that are being reduced by a certain factor all the way until infinity, you will eventually reach that whole number integer, after adding an infinite number of terms.