The easiest way of seeing that the answer is 2/9 is to reason as follows.
Consider .111111… If we multiply it by 9, we get .99999…., which is 1, so .11111… = 1/9, and .222222…. must be twice that value, or 2/9.
Suppose we had .123123123123123… . We can use similar reasoning to express the value as a fraction. Consider .001001001001001…. If we multiply this by 999, we get .99999…, which again is 1, so .0001001001001… is 1/999, and .123123123… is 123 times greater, and so equals 123/999.