OP wants a formal sense in which 0.333... and 1/3 are equal. Here is one.
One definition for equality between x and y is as follows: for all e greater than zero, |x - y| < e. Why is this a definition? (1) If x and y are equal, e.g., x=1 and y=1, this clearly holds. (2) If this holds, note that if x =\= y, there would be some e > 0 where the statement failed, so x must equal y. Thus if x and y are equal, this statement holds (1); and if the statement holds, x and y are equal (2).
Now we'll show x = 0.333... and y = 1/3 satisfy this definition. Pick any e > 0. A little informally, it's clear that for some decimal place, z = 0.3333...333000... has |z - 1/3| < e. Because z < x, |x - 1/3| < |z - 1/3|. Thus, |x - 1/3| < e.
(The answer earlier using a geometric series works too, but that the series equals a/(1-r) actually relies on the logic expressed above.)