Not sure what your question is.
You need the definition of a function to prove that it is onto, don't you? You can't prove that some unknown function is a surjection because not all functions are.But, what is the purpose of the rule h(n) = 4n-1 ?
How does it follow that h is onto if its values are always integers? For h to be onto, its values must be all integers.So on, and in this case 4n-1 will always be an integer thus h will become onto.