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REAL ANALYSIS NOTES

1. Background
1.1. Sets.
Definition 1. Two nonempty sets A and B have the same cardinality if there is a function f : A → B that
is both one-to-one and onto. In this case we write A ∼ B.
Definition 2. A nonempty set A is countable if either A is finite or A has the same cardinality as N. In
the latter case, A is countably infinite.
Lemma 3 (The Pidgeonhole Principle). Let f : A → B be any function, where A and B are finite sets,
and B has fewer members than A. Then f is not one-to-one.
Lemma 4. If A is a finite set and B ⊂ A, then B is finite too.
Lemma 5. If A is and B are finite sets, then A ∪ B, A ∩ B, A \ B, and A × B are all finite.
Lemma 6. Every finite set of real numbers contains a maximum and a minimum element.
Proposition 7. Let S be countably infinite, and let T ⊂ S be any nonempty subset. Then T is either finite
or countably infinite.
Lemma 8. The cartesian product of countable sets is countable. The union of countable sets is also countable.
Corollary 9. The set Q of rational numbers is countable.
Theorem 10. R is uncountable.
Corollary 11. The interval (0, 1) is uncountable. Every nonempty open interval (a, b) is uncountable. The
irrational numbers are uncountable.

1.2. Absolute Values.
Theorem 12. Let x and y denote arbitrary real numbers; assume k > 0.
(1)
(2)
(3)
(4)
(5)

−|x| ≤ x ≤ |x|.
|x| = | − x|; |x − y| = |y − x|.
|x| < k ⇐⇒ −k < x < k.
|x| > k ⇐⇒ x > k or x < −k.
|x · y| = |x| · |y|.

Theorem 13 (The Triangle Inequality ). Let x and y be any real numbers. Then |x + y| ≤ |x| + |y|.
Variations:
(1) |x − y| ≥ ||x| − |y||
(2) |x| − |y| ≤ |x − y| ≤ |x| + |y|.

Date: Spring 2013.
1

1.3. Bounds.
Definition 14. Let S be a nonempty set of real numbers.
(1) S is bounded above if there exists a number M with s ≤ M for all s ∈ S.
(2) S is bounded below if there exists a number m with s ≥ m for all s ∈ S.
(3) S is bounded if it is bounded both above and below.
Lemma 15. Let S ⊂ R be a nonempty set. S is bounded if and only if there exists M > 0 such that |s| ≤ M
for all s ∈ S. Properties:
(1) If S ⊂ T and T is bounded, then S is bounded, too.
(2) If S is finite, then S is bounded.
(3) Let |S| = {|s| | s ∈ S}. Then S is bounded if and only if |S| is bounded.
(4) Let S + T = {s + t | s ∈ S, t ∈ T } and ST = {st | s ∈ S, t ∈ T }. If S and T are bounded, then so are
S + T and ST .
Definition 16. Let S ⊂ R be a nonempty set. A number α is the infimum of S, and we write α = inf(S)
(or α = glb(S)) if
(1) α is a lower bound for S; and
(2) if α0 is any lower bound for S, then α ≥ α0 .
A number β is the supremum of S, and we write β = sup(S) (or β = lub(S)) if similar inequalities hold for
upper bounds.
Definition 17. If a is a number and S ⊂ R is a nonempty set, then S is bounded away from a if, for some
δ > 0, |s − a| > δ for all s ∈ S.
Definition 18. Let f be a real-valued function and A a subset of the domain of f .
(1) If f (a) ≤ M for all a ∈ A, then f is bounded above by M on A.
(2) If f (a) ≥ m for all a ∈ A, then f is bounded below by m on A.
(3) If f is bounded above and below on A, then f is bounded on A.

1.4. Completeness.
Definition 19 (The Completeness Axiom for R). Every nonempty set of real numbers that is bounded
above has a supremum. In symbols: If S ⊂ R, S 6= ∅, and S is bounded above, then there is a real number
β such that β = sup(S).
Theorem 20 (Order Properties of Numbers). The following are useful properties of the real numbers
as an ordered set.
(1) N is unbounded: Given any real number M , there exists a positive integer n with n > M .
(2) Squeezing in: For every positive number , no matter how small, ther exists a positive integer n such
that 0 < n1 < .
(3) The Archimedian principle: Given real numbers a and b with 0 < a < b, there exists a positive
integer n with na > b.
Theorem 21 (The Nested Intervals Theorem). Consider a nested infinite collection
I1 ⊇ I2 ⊇ I3 ⊇ I4 ⊇ . . .
of closed and bounded intervals. The intersection
I1 ∩ I2 ∩ I3 ∩ I4 ∩ . . .
contains at least one point. If the intervals’ lengths shrink to zero, then the intersection is a single point.
Theorem 22. Let x and y be any two real numbers, with x > y.
(1) The interval (x, y) contains at least one rational and one irrational number.
(2) The interval (x, y) contains infinitely many rationals and infinitely many irrationals.
2

2. Sequences and Series
2.1. Sequences and Convergence.
Definition 23. For every > 0 there exists a natural number N so that |an − L| < whenever n > N .
Definition 24. Let {an } be a sequence of real numbers.
(1) Boundedness: {an } is bounded above if there exists M such that an ≤ M for all n ∈ N; it is bounded
below if there exists m such that an ≥ m for all n.
(2) Monotonicity: {an } is increasing if an ≤ an+1 for all n ∈ N; {an } is decreasing if an ≥ an+1 for all
n ∈ N; {an } is monotone if it is either increasing or decreasing.
(3) Strictness: {an } is strictly increasing if an < an+1 for all n ∈ N. {an } is strictly decreasing if
an > an+1 for all n ∈ N
Theorem 25 (Monotone Convergence Theorem). If a sequence {an } is (i) monotone, and (ii) bounded,
then {an } converges.
Theorem 26. If a sequence {an } converges, then {an } is bounded both above and below.
2.2. Working with Sequences.
Theorem 27 (Limit Theorem of Sequences). Let {an } and {bn } be convergent sequences. Let a, b, and
c be real numbers, with an → a and bn → b.
(1) Sums and differences: The sequence {an ± bn } converges to a ± b.
(2) Constant multiples: The sequence {can } converges to ca.
(3) Products: The sequence {an · bn } converges to a · b.
(4) Quotients: If b 6= 0 and {bn } =
6 0 for all n, then the sequence { abnn } converges to ab .
Theorem 28 (The Squeeze Principle). Let {an }, {bn }, and {cn } be sequences such that an ≤ bn ≤ cn
for all n. If an → L and cn → L, then bn → L, too.
Proposition 29. Let {an }, {bn }, and {cn } be sequences and L a number.
(1) xn → 0 if and only if |xn | → 0
(2) If an → L, then |an | → |L|
(3) If |an | → 0 and {bn } is bounded, then an bn → 0.
Lemma 30. Let L be any number. There exist sequences {rn } and {pn }, both converging to L, with rn ∈ Q
and pn 6∈ Q for all n. If desired, {rn } and {pn } can be chosen to be either strictly increasing or strictly
decreasing.
Definition 31 (diverging to infinity). A sequence {xn } diverges to infinity if for every every M > 0 there
eists a number N such that xn > M whenever n > N .
2.3. Subsequences.
Definition 32. Let {an } be a sequence, regarded as a function a : N → R. A subsequence {ank } of {an } is
a composite function a ◦ n : N → R, where n : N → N is any strictly increasing function.
Theorem 33. Let {xn } be a sequence and L a number.
(1) If {xn } converges to L, then every subsequence {xnk } converges to L, too.
(2) If {xn } diverges to ±∞, then every subsequence {xnk } diverges to ±∞, too.
(3) If {xn } has subsequences converging to different limits, then {xn } diverges.
Proposition 34. Every sequence has a monotone subsequence.
Lemma 35. Every unbounded sequence {xn } has a monotone subsequence that diverges to ±∞.
3

Lemma 36. Let {xn } be a bounded sequence, with infimum α and supremum β. If α 6∈ {xn }, then there
is a decreasing subsequence {xnk } with xnk → α. If β 6∈ {xn }, then there is a increasing subsequence {xnk }
with xnk → β.
Theorem 37 (Bolzano-Weierstrass Theorem). Every bounded sequence {xn } has a convergent subsequence.

2.4. Cauchy Sequences.
Definition 38 (Cauchy Criterion). A sequence {xn } is Cauchy if, for every > 0, there exists N such
that |xn − xm | < whenever n > m > N .
Proposition 39. If {xn } converges, then {xn } is Cauchy.
Proposition 40. If {xn } is Cauchy, then {xn } is bounded.
Theorem 41. A sequence {xn } converges if and only if {xn } is Cauchy.

4

Quick Formal Definitions
Cardinality. ∃ non-empty sets A and B and a bijective function f : A → B. Written A ∼ B.
One-to-one. ∀x, y ∈ A, if f (x) = f (y), then x = y.
Onto. ∀y ∈ B, ∃x ∈ A such that f (x) = y.
Finite. ∃ a set A such that A = ∅ or A ∼ {1, 2, . . . , n} for some positive integer n. Otherwise, A is infinite.
Countability. ∃ a set A that is either finite or maps to the integers (countably infinite).
The Pidgeonhole Principle. ∃f : A → B, where A and B are finite sets, such that |B| < |A|. Then f is
not one-to-one.
The Triangle Inequalities. Let x and y be any real numbers. The following are true.
|x + y| ≤ |x| + |y|
|x − y| ≥ ||x| − |y||
|x − y| = |x + (−y)| ≤ |x| + |y|
|x − y| = |x − a + a − y| ≤ |x − a| + |a − y| = |x − a| + |y − a|
|x| − |y| ≤ |x − y| ≤ |x| + |y|.
Boundedness of sets.
Bounded above: ∃M such that ∀s ∈ S, s ≤ M .
Bounded below: ∃m such that ∀s ∈ S, s ≥ m.
Bounded: If S is bounded both above and below.
Bounded (alternative): ∃M > 0 such that ∀s ∈ S, |s| ≤ M .
Supremum. The supremum (or least upper bound ) of a set S ⊂ R which is bounded above is an upper
bound β ∈ R of S such that β ≤ u for any upper bound u of S.
Alternate definition. An upper bound β of a set S ⊂ R is the supremum of S if and only if for any > 0
there exists an s ∈ S such that β − < s.
Infimum. The infimum (or greatest lower bound ) of a set S ⊂ R which is bounded below is a lower bound
α ∈ R of S such that α ≤ l for any lower bound l of S.
Alternate definition. A lower bound α of a set S ⊂ R is the infimum of S if and only if for any > 0
there exists an s ∈ S such that s < α + .
Bounded away from a set. If a is a number and S ⊂ R is a nonempty set, then S is bounded away from
a if, for some δ > 0, |s − a| > δ for all s ∈ S.
Boundedness of functions. Let f be a real-valued function and A a subset of the domain of f .
Bounded above: If f (a) ≤ M for all a ∈ A, then f is bounded above by M on A.
Bounded above: If f (a) ≥ m for all a ∈ A, then f is bounded below by m on A.
Bounded: If f is bounded above and below on A, then f is bounded on A.
The Completeness Axiom for R. If S ⊂ R, S 6= ∅, and S is bounded above, then ∃β ∈ R such that
β = sup(S).
Unboundedness. ∀M ∈ R, ∃n ∈ Z+ with n > M .
Squeezing in. ∀ > 0, ∃n ∈ Z+ such that 0 < n1 < .
Archimedean principle. ∀a, b ∈ R with 0 < a < b, ∃n ∈ Z+ with na > b.
Convergence. ∀ > 0, ∃N ∈ N such that n > N implies |an − L| < .
The Squeeze Theorem. Let {an }, {bn }, and {cn } be sequences such that an ≤ bn ≤ cn for all n. If
an → L and cn → L, then bn → L.
Monotone Convergence Theorem. If a sequence {an } is (i) monotone, and (ii) bounded, then {an }
converges.
Comparison test for Squeezing. If an ≤ bn for all n and an → ∞, then bn → ∞.
Bolzano-Weierstrass Theorem. Every bounded sequence {xn } has a convergent subsequence.
Cauchy Criterion. ∀ > 0, ∃N ∈ N such that (∀n)(∀m) n > m > N =⇒ |xn − xm | < .
Cauchy and Convergence. Let {xn } be a sequence.
If {xn } converges, then {xn } is Cauchy.
If {xn } is Cauchy, then {xn } is bounded.
A sequence {xn } converges if and only if {xn } is Cauchy.

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