Assuming that by "any" you mean "any particular", you would have to read all the lines first. Otherwise, if the code selects the first line with probability 1/K, then I can just input some other number of lines.
I'm not sure if I understood your post correctly, but it seems to me that you didn't read the code carefully. The code selects the last read line with probability 1/n, where n is the number of strings already read, not the number of all strings. Say you have have already read n strings and they are all selected with probability 1/n. Now you read a new string and select it with probability 1 / (n+1). The probability that you didn't select the new string is then n / (n + 1) and the first n strings are selected with probability 1 / n * n / (n + 1) = 1 / (n + 1), so you now have n+1 lines all selected with probability 1 / (n + 1). So if the property that input strings are all selected with equal probability holds for n, it must also hold for n+1. Since it obviously holds for n = 1 (we just select the only string with probability 1 in that case), it must hold for all n.
