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> We can repeat this process for an infinite amount of time and still not have an infinite number of nines.

Isn't this somewhat ill-defined? You may have a finite number of 9s at any one point when adding more 9s, but there is no point "at infinity" where you can stop and look at how many 9s you've added because by definition there are still more 9s to add.

It's like trying to prove infinity is impossible:

> Assume that we have a finite number. Add 1 to that number. The result is not infinite. Add another 1. The result is still not infinite. We can repeat this process for an infinite amount of time and the number is still not infinity.

Sure, individual numbers "on the way" to infinity are not infinite, but that doesn't necessarily disprove the existence of infinity.

> We can repeat this process for an infinite amount of time

Well, no, you cant. How about I prove it to you:

Assume that we have a finite number of processes. Repeat the process. The number of processes is still not infinite.

/s

I tried this route as a counter to Cantor's diagonal argument, and got chastised by my then professor. I hope you have better luck, as I was never able to convince myself otherwise.

By that point you were already working in a universe where infinity was in some sense real. The universe of mathematics is different than the universe we live in.

> How about we prove that an infinite number of 9s is impossible?

First you need to define what you even mean by this statement. The rest of what you wrote makes no sense either.