I want to give some examples of showing numbers are equal by showing the difference between them is ε. It’s a fairly abstruse idea but when it works amazing things become possible.
The easy example, although one that produces strong resistance, is showing that the number 1 is equal to the number 0.9999…. But here I have to say what I mean by that second number. It’s obvious to me that I mean a number formed by putting a decimal point up, and then filling in a ‘9’ to every digit past the decimal, repeating forever and ever without end. That’s a description so easy to grasp it looks obvious. I can give a more precise, less intuitively obvious, description, though, which makes it easier to prove what I’m going to be claiming.
Another way to think of the number 0.9999…, with an unending series of 9’s ever after the decimal, is to think of making a sum. It’s the sum of the number 9 times (1/10), plus 9 times (1/100), plus 9 times (1/1000), plus 9 times (1/10,000), and so on and so on, adding 9 times the reciprocal of every whole power of ten. 9 times 1/1,000,000 is in that summation somewhere; so is 9 times 1/1,000,000,000,000, and 9 times 1 over a googolplex, and 9 times 1 over a one followed by Skewes’s number of zeroes, and so on. Remarkably, this sum of infinitely many terms never exceeds a finite number — in fact, it never exceeds the number 1 — in a wondrous property known as “convergence”.
But does it equal 1? It seems to me people divide their reactions instinctively along two lines. Some say obviously, this number has to equal 1. Others say no, there’s some difference, some gap between the number 0.9999… and 1. Most will, given a reasonable argument, come to concede that despite the gut feeling the two have to be the same number.
I want to prove 1 and 0.9999… are the same number. So I have to prove the difference between them is smaller than every possible choice of ε that’s a positive number. I’ll do that with a couple of examples first for specific choices of ε, and that will show the way to prove it for every choice.
Let’s pick an ε. Say, 0.01. Is the difference between 1 and 0.9999… smaller than this ε of 0.01? Well, 0.999 is certainly smaller than 0.9999…, so, the difference between 1 and 0.9999… is smaller than the difference between 1 and 0.999. And the difference between 1 and 0.999 we can tell from ordinary subtraction, and is 0.001, which is smaller than 0.01. Therefore — this is through a trick known as the transitive property — the difference between 1 and 0.9999… is smaller than 0.01, even if we haven’t said just what it is.
Let’s try a different ε, say, 0.0001. Is the difference between 1 and 0.9999… smaller than this ε of 0.0001? We can imitate the above argument. 0.99999 is smaller than 0.9999…, so, the difference between 1 and 0.9999…. is smaller than the difference between 1 and 0.99999. The difference between 1 and 0.99999 is 0.00001, which is definitely smaller than our ε of 0.0001. So the difference between 1 and 0.9999… is smaller than our chosen ε of 0.0001, even though we again don’t know what it is.
How about if ε were 0.025? It doesn’t always have to be some long series of zeroes terminated by a 1; in fact, it can’t always be. Like the above, though, 0.99 is less than 0.9999…, and the difference between 1 and 0.99 is 0.01, which is less than the ε of 0.025, so the difference between 1 and 0.9999… is smaller than this ε.
You can carry on trying this out until you get tired. I admit I’m getting a little worn out of it myself. But we have a routine for a proof here that’s going to work well enough. Given an ε by some doubter we look for the biggest power of one-tenth — zeroes after the decimal, followed by a 1 — that’s less than ε. Count up the number of digits are written there. Write down just as long a string of 9’s after the decimal and stop there. This is a number definitely less than 0.9999…, so that the difference between 1 and 0.9999… is less than the difference between 1 and our new number. But the difference between 1 and our new number is that string of zeroes followed by a 1, which is definitely less than ε. So, whatever ε we picked to start with, as long as it’s a positive number, we know that the difference between 1 and 0.9999… is less than that ε.
Since the difference between 1 and that 0.9999… is smaller than every possible pick of positive number, it follows that the difference has to be zero. And if the difference between two numbers is zero, they must be the same number. And what do you think of that?