Q:

Is the mapping from bits to ints one to one and onto? 

Is the mapping of the set of binary numbers (defined as character sequences, e.g. $0, 1, 10, 11, ....$) to the set of non-negative integers (e.g. $0, 1, 2, 3...$) one to one and onto?   
It's clear that $0$ in binary maps to $0$ in the integers, $1$ in binary maps to $1$ in the set of integers, and $10$ in binary maps to the integer $2$, and so on, but does this trend of one to one and onto mapping hold as both sets approach infinity?   

A:

HINT
To prove this mathematically, you should mathematically define the mapping you are talking about.  So, define the function $f:B \mapsto \mathbb{N}$ from the set of binary strings $B$ to the set of natural numbers (non-negative integers) $\mathbb{N}$ as follows:
Where $b$ is a bitstring $b_n b_{n-1} ... b_1 b_0$:
$$f(b)= \sum_{i=0}^n b_i \cdot 2^i$$
With this, you should be able to rigorously prove that $f$ is injective and surjective.