Say you're trying to manually find the square root of 2

You know it's between 1 and 2, so 1 must be the first digit of the square root

You square every number between 1 and 2 at increments of 1/10, and 1.4 is the largest number who's square is below 2. 1.4^2 = 1.96, so 1.4 must be the first two digits

Then you square every number between 1.4 and 1.5 at increments of 1/100, and 1.41 is the largest possible approximation. 1.41^2 = 1.9881, 1.41 must be the first 3 digits

Between 1.41 and 1.42 at increments of 1/1000, re1.414 = 1.999396

Between 1.414 and 1.415 at increments of 1/10000, 1.4142^2 = 1.99996164

As you keep going, the square of your approximation will approach 2, but never exactly 2. It will always be 1 and a number of 9s after the decimal point

So you might say if you do this an infinite number of times, the square of the limit will be 1.999..., but that can't be right because we were trying to find the square root of 2. So this means 1.999... = 2

And if you subtract 1 from both sides, it's 0.999... = 1

Therefore, 0.999... = 1