Any real number in [0,1] has a unique binary decimal representation o.b_{1}b_{2}b_{3}..., where each b_{i}is either 0 or1. Numerically, o.b_{1}b_{2}b_{3}...=from1 to ∞ Σ(b_{n}/2^{n})

where the infinite series converge to a number in[0,1].

The question is- how does it possible for this series "to converge to a number in[0,1]" for example, how 1/3 can be expressed using this series? Is there a proof for that statement?

Thanks in advance