But clearly, if we add up all the infinite fractions between 0 and 1, they would add up to 1.
0.9 and 0.8 are in that set, so they would add up to at least 1.7. In fact if you give me any positive number I can give you a (finite!) set of (distinct) fractions less than 1 which sum to more than that number. In other words, the sum is infinite.