Set of measure zero-can't pinpoint my mistake

I think I am being confused by the definition of this(in a differential geometry sense) and was curious what I was getting wrong. The definition I have seen is that a set has measure zero if for any delta>0 there exists a countable cover of the set by open cubes s.t. the volume of the cubes is less than delta. Now, I have run into a strange conclusion from this; If I take a countable cover of R^1 by taking an interval of positive length around every rational number(does this work? I think it should as Q is dense), we can choose intervals of length (delta/2)*(1/2^n) for the nth rational number, we can sum these intervals to get a cover of R^1 with size delta/2<delta, which implies R^1 has measure zero in R^1. I know that this is very very wrong. However, I'm failing to see where in my thought process I made an error. Please help me find my mistake.

Thanks