How do I prove that $x_0 = \frac{\lceil 2^n x \rceil - 1}{2^n}$ can be represented with a finite fractional binary representation with $n$ bits?
The definition of finite fractional binary representation is that a number $x$ can be represented by $x = (0.b_1 b_2 ... b_k)_2.$
I have tried drawing a picture and I am going nowhere? Can anyone help me with this please?
Just like moving the decimal point in base 10 is equivelent to multiplying or dividing by powers of 10, in binary shifting it is equivalent to multiplying or dividing by powers of 2.
Thus you have a numerator which is an integer of finite length base 2 and a denominator which acts to shift the decimal point, keeping it nonetheless finite.