Thank you but I am not having trouble with the concept of infinity or infinities, and I understand that the integers are a subset of the real numbers. In fact, the relationship of integers to real numbers can be used to illustrate that there is more than one infinity. We all agree that there are an infinite number of integers, right? And yet for any two consecutive integers such 2 and 3 (or any two integers for that matter), there is an infinite number of real numbers that can be represented in decimal form between them, such as 2.9, 2.99, 2.998, 2.999... So the infinite number of real numbers must be greater than the infinite number of integers.

All I am arguing is that there are subtle mathematical differences between two numbers if one of them a tiny bit larger than 3 and the other is an equally tiny bit smaller than three. I can easily imagine 2.99999... I can just as easily imagine a number which is greater than three by the same degree. They are all three points on a number line, with the integer 3 in the middle and the two repeating-decimal numbers flanking it, equidistant on either side. The fact that the difference between them is essentially unmeasurably small doesn't change the fact that they are different.

To put it another way, take the limit of x as x approaches 3 from a number greater than three, and then take the limit of x as x approaches 3 from a value less than three. You plug those two values into f(x)=3/(x-3) and your answers will converge on to two different values. In fact, the answers converge on infinity, the first one with a positive sign and the other with a negative sign. That's because in the first case the denominator remains positive, because you are subtracting a number less than 3 from 3. As x converges on its limit of 3 from a number greater than three, (x-3) converges on zero, but remains greater than zero. The casual observer can see that for a number converging on 3 from say, 2, (x-3) remains a negative number that converges on zero "from below."

So my argument is that if 2.9999... is the same as 3, then (4-0.999999999...) is the same as three. But they are not as f(x)=3/(x-3) shows. What I am arguing more or less is that 2.9999... converges on 3 but is not quite the same as 3. Now maybe there is some flaw in my reasoning, but nobody has pointed it out yet to my satisfaction.