This works in any simple base using the same concept of decimal representation.

In base x, consider the number zero followed by n digits of (x-1) after the decimal point, e.g. 0.FF...F with n Fs for hexadecimal. One minus this number gives the difference 1/x^n. In the limit n goes to infinity, this difference goes to zero for real numbers. And with the real numbers, zero difference means they are the same number.

This is only true if you define limit "math" to mean arithmetic and what simple calculators do. Algebra gives the abstract tools to work with numbers without needing the decimal expansion. By trigonometry and especially calculus, pi gets used a lot in an exact sense. Although sometimes the fundamental basis of what it means to work with an real numbers doesn't get covered until a course on real analysis.

Mathematical proofs are a way of finding new properties of a system by making deductions from previously known properties, and in a practical sense are often a short-cut finding, the new property without testing every possible case.

For a simple example, consider the property that every integer multiplied by by 10 will end up with a zero in the ones place. Someone could respond: "How could you know that? There are an infinite number of integers and it would take infinite amount of time to multiply each by 10 to check it." But using a proof can rigorously show this is a pattern without testing every number by exploiting the properties of numbers.

In the case of multiplying 0.999... you can workout what the pattern any given digit will follow, and use that instead of manually performing the calculation.