1. Let x ∈ . Prove that x = sup {q ∈ Q : q 2. Prove Euclid’s division algorithm: If a and b are natural numbers, then there exist unique numbers q and r, each of which is either 0 or a natural number,...

1. Let x ∈
. Prove that x = sup {q ∈ Q : q

2. Prove Euclid’s division algorithm: If a and b are natural numbers, then there exist unique numbers q and r, each of which is either 0 or a natural number, such that r <>

3. Let S ⊆ R. Mark each statement True or False. Justify each answer.

(a) int S ∩ bd S = ∅

(b) int S ⊆ S

(c) bd S ⊆ S

(d) S is open iff S = int S.

(e) S is closed iff S = bd S.

(f ) If x ∈ S, then x ∈ int S or x ∈ bd S.

(g) Every neighborhood is an open set.

(h) The union of any collection of open sets is open.

( i ) The union of any collection of closed sets is closed.