Let E be compact and nonempty. Prove that E is bounded and that sup E and inf E both belong to E.

For the first part can we use the fact that if E is compact iff it is closed and bounded? So since its compact than its closed and bounded..or is do I have to actually prove that its bounded? And also for the sup and inf, i'm not sure but would we set E=the interval and show that they are the inf and sup?