5 The Symbols ±∞
Thus far the only subsets of the real numbers that have a supremum are those which are non-empty
and bounded above. In this very short section, we introduce the ∞-symbol to provide all subsets of the
real numbers with both a supremum and an infimum.
Definition 5.1. Let S ⊆ R be any subset. If S is bounded above/below, then sup S/inf S are as in
Definition 4.6. Otherwise:
1. We write sup S = ∞ if S is unbounded above, that is
∀x ∈ R, ∃s ∈ S such that s > x
2. We write inf S = −∞ if S is unbounded below,
∀y ∈ R, ∃t ∈ S such that t < y
3. By convention, sup ∅ := −∞ and inf ∅ := ∞, though these will rarely be of use to us.
The symbols ±∞ have no other meaning (as yet): in particular, they are not numbers! If one is willing
to abuse notation and write x < ∞ and y > −∞ for any real numbers x, y, then the conclusions of
Lemma 4.8 are precisely statements 1 & 2 in the above definition!
Examples 5.2. 1. sup R = sup Q = sup Z = sup N = ∞, since all are unbounded above. We also
have inf R = inf Q = inf Z = −∞ (recall that inf N = min N = 1).
2. If a < b, then any interval [a, b], (a, b), [a, b) or (a, b] has supremum b and infimum a, even if one
end is infinite. For example,
S = (7, ∞) = {x ∈ R : x > 7}
has sup S = ∞ and inf S = 7.
3. Let S = {x ∈ R : x
3
−4x < 0}. With a little factorization, we see that
x
3
−4x = x(x −2)(x + 2) < 0 ⇐⇒ x < −2 or 0 < x < 2
It follows that S = (−∞, −2) ∪ (0, 2), from which sup S = 2 and inf S = −∞.
Exercises 5. 1. Give the infimum and supremum of each of the following sets:
(a) {x ∈ R : x < 0} (b) {x ∈ R : x
3
≤ 8}
(c) {x
2
: x ∈ R} (d) {x ∈ R : x
2
< 8}
2. Let S ⊆ R be non-empty, and let −S = {−s : s ∈ S}. Prove that inf S = −sup(−S).
3. Let S, T ⊆ R be non-empty such that S ⊆ T. Prove that inf T ≤ inf S ≤ sup S ≤ sup T.
4. If sup S < inf S, what can you say about S?
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