Please critique my proof on the fact that the algorithm for decimal to binary conversion always works.

I feel like I know where I want to go with this, but I am not very good at putting it into words.

Prove that the decimal to binary conversion method covered in class always works. Proof: Note that if a decimal number can be represented in the form N=bn2n + bn-12n-1 + ... +bo2 where b is either 0 or 1, then its binary representation would be bnbn-1.....bibo. We need to show that the algorithm for converting a number from decimal to binary form will always result in a number of this form. Since every term other than bo is a multiple of 2, we can rewrite the above form as N= 2(bn2n-1 + bn-12n-2 + ... + bi) + bo. This means that when we divide N by 2, we get a quotient of (bn2n-1 + bn-12n-2 + ... + b) and a remainder of bo. The remainder bo will be either 0 or 1, depending if N is even or odd, and will be the right-most digit of the binary representation of N. Note that we can continue to factor 2 out of the quotient such as: (bn2n-1 + bn-12n-2 + ... + b1)= 2(bn2n-2 + bn-1 2n-3 + ... + b2) + bi where bi is the remainder of the first quotient divided by two. We can see now that our binary digits emerge by repeating the process of division by 2. We can continue to repeat this process until we reach a quotient of 0, and then collect the remainders backwards to reach our binary representation of bnbn-1......bibo. Therefore, it is sufficient to conclude that any decimal number can be expressed in binary form using the algorithm discussed in class