Why do i have to show that a problem L $ \in $ NP before i start a polynomial reduction? i want to do a polynomial reduction from the Independent-Set-Problem which is NP-complete to the AUCTION-Problem, to show that AUCTION $\in$ NP-complete,
but why do i always have to show first, that our problem(in this case AUCTION) $ \in $ NP?
Because a problem L is NP-complete if and only if:


L is in NP
L is NP-hard: every problem A in NP can be polynomially reduced to L.


So to show that L is NP-complete, you have to demonstrate both the properties.