Determine a number in interval with binary representation

Question

Response to the following problem:

Given a number a in the interval [0, 1) with an n-h ii binary representation [b₁, b₂ ........... bn] show that for any other number b to have a binary representation with [b₁, b₂ ........... bn] as the prefix, b has to lie in the interval [a,a +1/2ⁿ] .

Make sure you use enough details to support your answer.