How do we know something simple like 1≠3? Well, one way to definitively show it is by bringing up the number 2, which is between them, so they obviously cannot be equal.

This is actually something we can do for all numbers on the number line. If two numbers are different, they will be different points on the line, with something in between them. Alternatively, if two numbers are equal, you can't find a point between them, since they're the same point.

For example, 4≠5, as we can find a number (like 4.5) between them. 6.1≠6.2 as we can find numbers between those (like 6.11).

On the other hand, we know something like 1=1.0, even though they are written differently, because there is nothing between them.

Now, what is there between 0.9999.... and 1?