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Exercise 1: Absolute value function

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exercise 1:
Using properties 1, 2, and 3 of the absolute value function on Q stated in class, show that
for all x, y in Q:
(i). if x 6= 0, |
1
x
| =
1
|x|
,
(ii) ||x| − |y|| ≤ |x − y| ≤ |x| + |y|.
exercise 2:
Let T = (0, 1) ∪ {2}. Find, with proof, sup T.
exercise 3:
Let S and T be two bounded above subsets of R. Define the subset
S + T = {x + y : x ∈ S, y ∈ T}.
Show that S + T is bounded above.
exercise 4:
From Abott’s textbook: exercise 1.4.4.
exercise 5:
Using the definition of convergent sequences show that bn =
1

n
converges to zero.
exercise 6:
Using the definition of convergent sequences show that any constant sequence is convergent.

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