This is not a paradox, but a fact. Infinitely cyclic decimal 0.999 ... and 1 are exactly equal, not infinitely close, but exactly the same. You can think that they are just two ways to write the same number.

The equality between them is directly determined by the construction process of real numbers, and the strict proof process can't avoid the two methods of constructing real numbers, namely, Dydykin division method and Cauchy sequence method, which are equivalent.

Law of integer division:

1) Start with the high order of the dividend, first look at how many digits there are in the dividend, and then try to divide the dividend by the first few digits. If it is less than the divisor, try dividing it by one digit.

2) Write the quotient except the dividend on the dividend.

3) The remainder after each division operation must be less than the divisor.

Mathematics and reality:

Mathematics can have nothing to do with reality, and its key is definition. Different definitions can make him equal or unequal.

If you stay on the definition of rational number (i.e. fraction) and assume that 0.9999 ... is rational number, then 0.9999 ... is converted into component number, which is 11,which is undoubtedly1.

If you stay on the definition of real numbers, assuming that 0.9999 ... is a real number, then there are no other real numbers ... and 1 between 0.9999, and they are equivalent whether they are converted into sequence representation or de-de-de division, so they are also equal.