2. (a) (8 points) Using the ordered eld axioms of R, show that for any x; y 2 R, we have
xy 1
2×2 + 1
2y2:
State what axioms you are using in each step.
(b) (2 points) Prove or disprove the following inequality: for any x; y; z 2 R, we have
xyz 1
2×2 + 1
2y2 + 1
2z2
3. (a) (9 points) Show that for S a subset of R the following are equivalent:
(i) For all x; y 2 R with x < y, there is an s 2 S so that x < s < y.
(ii) For all x 2 R and for all > 0, there is an s 2 S so that jx sj < .
(b) (1 point) Does such a subset S of R satisfying (i) or (ii) exist? Explain you answer.
4. (a) (5 points) For each n 2 N, let In = [an; bn] = fx 2 R : an x bng be intervals
of R which are nested: In+1 In for every n 2 N. Show that the resulting nested
sequence of intervals
I1 I2 I3 I4
has a non-empty intersection; that is, 1
n=1In 6= ;.
(b) (5 points) Show that
1
n=1
0;
1
n
= ;:
This shows that we need intervals which include the endpoints for the claim in part
(a) to hold.
5. Let a1; b1 be two real numbers such that 0 < a1 < b1. For n 2 N, we dene
an+1 =
p
anbn and bn+1 =
an + bn
2
:
(a) (2 points) Use question 2 (a) to show that 0 < an bn for every n 2 N.
(b) (3 points) Show that fangn2N is increasing and fbngn2N is decreasing.
(c) (2 points) Show that the sequences fangn2N and fbngn2N are bounded.
(d) (3 points) Deduce that the two sequences converge and prove that they converge
to the same limit.
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