Here’s one more way of convincing students that . Here’s the idea: how far apart are the two numbers?
First off, since , we know that .
Of course, we know that . Since must lie between and , we know that must be less than .
Second, we know that . Since must lie between and , we know that must be less than .
Third, we know that . Since must lie between and , we know that must be less than .
By the same reasoning, we conclude that
for every integer . What’s the only number that’s greater than or equal to and less than every decimal of the form ? Clearly, the only such number is . Therefore,
, or .
I like this approach because it really gets at the heart of the difference between integers and real numbers . For integers, there is always an integer to the immediate left and to the immediate right. In other words, if you give me any integer (say, ), I can tell you the largest integer that’s less than your number (in our example, ) and the smallest integer that’s bigger than your number ().
Real numbers, however, do not have this property. There is no real number to the immediate right of . This is easy to prove by contradiction. Suppose is the real number to the immediate left of . That means that there are no real numbers between and . However, is bigger than and less than , providing the contradiction.
(For what it’s worth, the above proof doesn’t apply to the set of integers since doesn’t have to be an integer.)
By the same logic — visually, you can imagine reflecting the number line across the point — there is no number to the immediate left of . So while would appear to be to the immediate left of , they are in reality the same point.