6. If E, SCR", such that ECSC cl(E), where cl(E) is the closure of E, then E is said to be dense in S. For example, the Q of rational numbers is dense in R. If E is dense in S and S is dense in T, prove that E is dense in T, too. 7. If each of E and S are dense in R" and if S is open in R", show that EnS is dense in Rn. 8. Let S, TCR". Recall that S compact if and only if every open cover of S in R has a finite subcover in Rn. (i) Suppose SCT. Show that S is compact in (R", || ||) if and only if, S is compact in the metric subspace (T, || ||T). (ii) If S is closed in R" and T is compact. Show that SOT is compact, too. 9. Show that arbitrary intersection of compact sets of R" is compact. 10. Show that a finite union of compact subsets of R" is compact.