Real numbers proofs
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A. show that if a < b
then a < 1/2(a+b) < b
b. i. if 0 < b < 1
show that
0 < b^2 < b < 1
ii. if 1 < b
show that
1 < b < b^2
c. show that abs(x-a) < E if an only if
a-E < x < a+E
where E= epsilon
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Solution Summary
This provides examples of several proofs regarding real numbers.
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a. Proof:
If a < b, then a - b < 0, then (a - b)/2 < 0
So we have a - (a + b)/2 = (a - b)/2 < 0, this implies that a < (a+b)/2.
(a + b)/2 - b = (a - b)/2 < ...
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