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.

MA 3831

# Exercise 1: Absolute value function

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