real numbers in base 2 - how the infinite series (Σ(b/2^n)) converge to a number in [0,1]?

Any real number in $[0,1]$ has a unique binary decimal representation 0.bbbbb, where each b is either $0$ or $1$. Numerically, $0.b_1 b_2 b_3 b_4 b_5...=\sum^{\infty}_{i=1}(b_n/2^n) $
(***b_1=b-sub-one) 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