I want to do some more tricky examples of using this ε idea, where I show two numbers have to be the same because the difference between them is smaller than every positive number. Before I do, I want to put out a problem where we can show two numbers are not the same, since I think that makes it easier to see why the proof works where it does. It’s easy to get hypnotized by the form of an argument, and to not notice that the result doesn’t actually hold, particularly if all you see are repetitions of proofs where things work out and don’t see cases of the proof being invalid.
So let me start again with the number 1, and then another number, something that’s got an infinite number of digits, and repeating digits at that. This second number I’ll make 0.99998888… That is, after the initial run of four 9’s after the decimal, it then becomes an endlessly repeating string of 8’s. This number is not equal to 1, but we can try using the ε-style proof to see if they are equal. If we are able to prove they’re equal then we have some interesting difficulties ahead. If we can prove they’re not equal we can maybe trust this ε thing more.
I do want to point out a bit of notation here: by the trailing … I want to say that the last digit gets repeated forever. The number is (9999/10000) plus 8/100,000 plus 8/1,000,000 plus 8/10,000,000 plus 8/100,000,000 plus 8/1,000,000,000 plus … and so on forever. I’ll have some other numbers, such as 0.99998, which don’t end in an ellipsis, and what I mean there is that the number comes to an end just where I stopped writing: that number was 99998/100000, and nothing more is added to it.
My old claim was that two numbers have to be the same if the difference between them is smaller than every positive number. So let’s pick some positive numbers. Try ε of 0.01, for example. 0.99998888… is some number that’s bigger than 0.999, obvious enough just at a glance. So the difference between 1 and 0.99998888… has to be less than the difference between 1 and 0.999. But the difference between 1 and 0.999 is 0.001, which is less than our first choice of ε here. So far, the numbers are, if not equal, at least not very different.
Let’s try a smaller difference, say, ε of 0.00025. 0.99998888… is larger than 0.9999, so the difference between 1 and 0.99998888… must be smaller than the difference between 1 and 0.9999. The difference between 1 and 0.9999 is 0.0001, which is less than the ε of 0.00025. So far we’re doing well in proving these numbers are actually equal.
But, now, suppose that ε is 0.00001. 0.99998888… is some number that has to be less than 0.99999. So the difference between 1 and 0.99998888… has to be larger than the difference between 1 and 0.99999, which is 0.00001, our ε. We’ve found that the difference between 1 and 0.99998888… is not smaller than every possible positive number.
This isn’t an artifact of where we cut off this approximation, either. If we use the fact that 0.99998888… has to be less than 0.9999889, we still know that the difference between 1 and 0.99998888… has to be bigger than the difference between 1 and 0.9999889, which is 0.0000111.
The difference between 1 and 0.99998888… isn’t smaller than every possible positive number which we might choose as ε, and so, the two numbers are not equal.