Open cover rationals proper subset of R? If I were to cover each rational number by a non-empty open interval, would their union always be R? It seems correct to me intuitively, but I am quite certain it is wrong. Thanks

If you enumerate the rationals as a sequence $x_1, x_2, \dots$, you can then take a sequence of open intervals $(x_1-\delta, x_1+\delta), (x_2-\delta/2, x_2+\delta/2), (x_3-\delta/4, x_3+\delta/4), \dots$ which gives an open cover for $\mathbb{Q}$ of total length $4\delta$, which can be made as small as you wish, by choosing $\delta$ sufficiently small.
