Suppose is finite and . Prove that .
My attempt: I try to prove it by using induction on
Base case: Suppose has 1 element, then
I don't think my base case is 'rigorous' enough, it feels there is something missing...
For induction step:
Suppose for having n elements, and ,then .
Now let has n+1 elements, let be an arbitrary element, then has n elements, then
But after that, I have no idea how to proceed at all. Please provide any hint necessary. Thanks in advance.