Prove If E is compact and F is closed then E F is compactSol

Prove: If E is compact and F is closed, then E F is compact.

Solution

We show this by proving that E F is both closed and bounded,

First, since E is compact, E is also closed.

The intersection of closed sets is closed so E F is closed.

We need only show that E F is also bounded.

However, since E is compact, it is bounded

there is an M > 0 such that |x| < M for all x E.

Since E F E we must have |x| < M for all x E F as well.

Therefore, E F is both closed and bounded, and thus compact