I myself do not...personally believe this as a mathematical "fact," but also realize how futile it is to argue against it. So like those many, it is impossible to convince me as well, after all, there is a reason this idea is so highly controversial.
The term mathematical fact might be a little vague. First, not every number is rational -- not every number can be represented as the quotient of integers. For example the width of a square whose area is 2 is not a rational number. That is, we need the full blown set of real numbers. Figuring out what the real numbers (really) look like is a hard challenge, and providing a description of them in set theory was a major challenge. There are two main approaches: Dedekind's cuts and Cauchy sequences. They produce the same set. Essentially, take a bounded sequence of rational numbers, and identify a "number" L with this sequence. The real numbers are the rational numbers with all these Ls. Thus in the construction of real numbers, we see that every real number is the limit of a sequence of rationals.
In other words, the real number "1" is by definition the limit of the sequence