Practice of exam 2

2. Closure is closed

Let . Show that

is closed. Show that is compact is bounded.
Firstly, we can prove is closed by contradiction: suppose there exists such that , we have two cases:

1. If , it leads to a contradiction with due to and .
2. If and , we can say and which leads to a contradiction with .
   Hence in both cases we can say is closed proved by contradiction.