MATH 3210 - SUMMER 2008 - ASSIGNMENT #3
Absolute value and Bounded sets.
(1) Use induction to prove that for any nreal numbers: a1,...,anwhere n≥2, n∈N,
we have:
|a1+a2+. . . an| ≤ |a1|+|a2|+...|an|
This is called the generalized triangle inequality.
(2) (a) Which real xsatisfy: |x−2|+|x+ 3|>10 (Hint: work out the following cases:
x≥2, −3≥x < 2, and x < −3). Draw the set of solutions on the real line.
(b) Solve: x+ 2 <|x2−4|. Draw the set of solutions on the real line.
(3) Find sup /inf of the following sets, prove your assertions:
(a) A= (−2,3)
(b) B={b|b∈Qand b < 10}
(c) C={4−1
n2|n∈N}
(d) Find just the inf of D=5 + n
2n2+3n+1
n∈N
(4) For each part, find an example or explain why none exist:
(a) a set Asuch that inf A= 0 and sup A= 15
(b) a set Asuch that inf A= 15 and sup A= 0
(c) a set Awhich is bounded below and not bounded above.
(d) a set Awhich is bounded above and not bounded below.
(e) a set Awhich is bounded below and which doesn’t have an infimum.
(f) a set Afor which ∃a∈Asuch that a < 3 but 3 is not an upper bound for A.
1
