The density of the rational numbers and the

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If x and y are real numbers, x < y , then there exists an irrational number z
such that x < z < y .
Exercises 1.3
1. True – False. Justify your answer by citing a theorem, giving a proof, or giving a counter-
example.
2. True – False. Justify your answer by citing a theorem, giving a proof, or giving a counter-
example.
3. Let S ⊆ R be non-empty and bounded above and let u = sup S . Prove that u ∈ S if and
only if u = max S .
4. (a) Let S ⊆ R be non-empty and bounded above and let u = sup S . Prove that u is unique.
(b) Prove that if each of m and n is a maximum of S , then m = n .
5. Let S ⊆ R and suppose that v = inf S . Prove that for any positive number , there is an
element s ∈ S such that v ≤ s < v + .
6. Prove that if x and y are real numbers with x < y , then there are infinitely rational
numbers in the interval [ x, y ].
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I.4. TOPOLOGY OF THE REALS
Definition 6. Let S ⊆ R . The set S
c
= { x ∈ R : x / ∈ S } is called the complement of S .
Definition 7. Let x ∈ R and let > 0 . An -neighborhood of x (often shortened to
“neighborhood of x ”) is the set
N ( x, ) = { y ∈ R : | y - x | < } .
The number is called the radius of N ( x, ) .
Note that an -neighborhood of a point x is the open interval ( x - , x + ) centered at x
with radius .
Definition 8. Let x ∈ R and let > 0 . A deleted -neighborhood of x (often shortened to
“deleted neighborhood of x ”) is the set
N
*
( x, ) = { y ∈ R : 0 < | y - x | < } .
A deleted -neighborhood of x is an -neighborhood of x with the point x removed;
N
*
( x, ) = ( x - , x ) ∪ ( x, x + ) .