Let x be an upper bound for A subset Real #'s.

Prove that if x element A, then x = sup(A).

Proof:

Let x be an upper bound for A subset Real #'s.

Assume x element A.

a <= x for all a element A by definition of upperbound.

Since x element A, x <= x.

Therefore, since x is an upper bound for A and x <= x for all upper bounds x of A, x = sup(A).

I was wondering if this was a valid proof.

If not could you help correct it please.

Thanks