Foundations of Analysis / MAT 3213.001
Final / May 5, 1998 / Instructor: D. Gokhman
Name: Pseudonym:
1. (10 pts.) Suppose δ>0. Prove that |x−2|<δ⇒|x2−4|<δ(δ+4).
2. (10 pts.) Find all x∈Rsuch that |2x|≤|x−2|.
3. (20 pts.) Suppose f:A→Band g:B→C.
(a) Prove that if fand gare 1-1, then g◦fis 1-1.
(b) Prove that if fis onto and g◦fis 1-1, then gis 1-1.
4. (20 pts.) Suppose Aand Bare nonempty bounded subsets of R. Prove that A∪B
is bounded below and inf (A∪B)=min{inf A, inf B}.
5. (20 pts.) Suppose Ais a nonempty bounded subset of R. Prove that inf A∈∂A
by showing that inf A∈A, but inf A6∈ A◦.
6. (20 pts.) Suppose Aand Bare closed subsets of R. Determine whether each of
the following statements is true and prove the statement or give a counterexample.
(a) (A∩B)◦=A◦∩B◦
(b) (A∪B)◦=A◦∪B◦
7. (20 pts.) Let A=n1/n:n∈Z+o. Determine sup A,infA,maxA,minA,A◦,
L(A)andA.IsAclosed in R?IsAopen in R?
8. (20 pts.) Suppose Ais a nonempty subset of R, which is bounded below, but does
not have a minimum. Prove that L (A)6=Ø.
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