How do I prove this function is bijective?
$$v(s,p)=2^{p-1}(2s-1). $$
The domain is the natural numbers and the codomain is also the natural numbers So I have to somehow show that every natural number can be written as a product of the power of 2 and an odd number.
The fundamental theorem of arithmetic implies that every number can be expressed as a product of a power of two and an odd number. (The power of two is the twos in the prime factorization and the odd number is the product of the odd primes.) It also says this factorization is unique, which implies that the function is injective.
(If you need to prove this “from scratch” then more information about what facts you have at your disposal are needed. In general you can prove this just like the proof of fundamental theorem of arithmetic is proved.)