Analysis Summary

Set theory

Rational numbers Q are fractions { ^a/_b | a, b ∈ Z}.
√2 is an irrational number.
Every Real number has a unique decimal expansion (up to the ambiguity of ending in ...000... or ...999...).
Rational numbers have terminating or recurring decimal expansions.
A set which can be put into one-one correspondence with N is called countable.
Countable sets: Z, N × N, Q, ...
The Cantor diagonalisation argument shows that R is uncountable.
Subsets of countable sets are either finite or countable.
A countable union of countable sets is countable.

The Real numbers

The set of Real numbers satisfies the properties:
I (Algebraic axiom) R is a field under + and . .
II (Order axioms) There is a relation > on R satisfying:
   (a)  for any a ∈ R, exactly one of 0 > a, a = 0, a > 0 is true (Trichotomy),
   (b)  if a, b ∈ R with a, b > 0 then a + b > 0 and a b > 0,
   (c)  if a, b, c ∈ R and a > b then a + c > b + c.
III (Completeness axiom) Any bounded set has a least upper bound.

From these one may deduce:
(The Archimedean property) If a > 0 in R then for some n ∈ N we have a > ¹/_n > 0.
There is a rational number between any two real numbers.

Convergence in the Real numbers

A sequence (a₁ , a₂ , a₃ , ... ) converges to a limit α if:
   Given ε > 0 there exists N ∈ N such that for n > N we have | a_n - α | < ε.

Properties of convergence

Any convergent sequence is bounded (above and below).
Arithmetic properties: If (a_n)→ α and (b_n)→ β are convergent sequences, then (a_n + b_n)→ α + β, (a_n - b_n)→ α - β, (a_nb_n)→ αβ, and, provided that b_n , β ≠ 0 , (a_n/b_n)→ ^α/_β.
A sequence (a_n) for which a_n+1 ≥ a_n for all n, is called monotonic increasing.
A bounded monotonic increasing sequence converges to its least upper bound.
(Bolzano-Weierstrass theorem) Any bounded sequence contains a convergent sub-sequence.
A sequence (a₁ , a₂ , a₃ , ... ) is called a Cauchy sequence if:
Given ε > 0 there exists N ∈ N such that for m, n > N we have | a_m - a_n| < ε.
Cauchy sequences in R are convergent.

Continuity for Real functions

A function f : R→ R is said to be continuous at p ∈ R if:
   Whenever (x_n) is a sequence converging to p, the sequence ( f (x_n) ) converges to f (p).
One may prove that this definition is equivalent to the ε-δ definition:
A function f : R→ R is said to be continuous at p ∈ R if:
   Given ε > 0 there is a δ > 0 such that if | x - p | < δ then | f (x) - f (p) | < ε.
A function which is continuous at every point of R is called a continuous function.
If f and g are functions continuous at p, so are f + g, f - g, fg, and (provided that g(p) ≠ 0)  f / g.
If f is continuous at p and g is continuous at f (p) then the composite function h(x) = g f (x) = g (f (x)) is also continuous at p.

Metric spaces

A metric space is a set X together with a function d from X × X to R satisfying the properties:
   (a)  d(x, y) ≥ 0 and d(x, y) = 0 if and only if x = y,
   (b)  d(x, y) = d(y, x)
   (c)  d(x, y) + d(y, z) ≥ d(x, z) (the triangle inequality).

The space R² is a metric space under the metrics:

d₂((x₁, y₁), (x₂, y₂)) = √((x₁ - x₂)²+ (y₁ - y₂)²) (The usual or Pythagorean metric)

d₁((x₁, y₁), (x₂, y₂)) = |x₁ - x₂| + |y₁ - y₂| (The taxicab metric)

d_∞((x₁, y₁), (x₂, y₂)) = max {|x₁ - x₂|, |y₁ - y₂|} (The supremum or uniform metric).

The space B[0, 1] of bounded functions on the interval [0, 1] is a metric space under the metric:

d_∞((x_i), (y_i)) = max {|x₁ - y₁|, |x₂ - y₂|, ... } (The uniform metric).

The space C[0, 1] of continuous functions on the interval [0, 1] is a metric space under the metrics:

d_∞(f, g) = lub {|f (x) - g(x)|  | x ∈ [0, 1]} (The uniform or supremum metric).

d₁(f, g) = |f (x) - g(x)| dx.

Convergence in a metric space

A sequence (x_n) of points in a metric space (X, d) converges to a point α in X
if the real sequence (d(x_n, α)) converges to 0 in R.
Equivalently: if given ε > 0 there exists N ∈ N such that for n > N we have d(x_n, a ) < ε.

Convergence in the metric d_∞ is called uniform convergence. It is possible for sequences in spaces of sequences or functions to converge in one metric but not in others.

A space is called complete if every Cauchy sequence has a limit. The space C[0, 1] is a complete space under the metric d_∞ but not under the metric d₁.

Further properties of continuous functions

The Weierstrass approximation theorem states that every continuous function on a closed bounded interval is the uniform limit of polynomial functions

The Intermediate Value theorem states that a function which is continuous on an interval and which is positive at some point and negative at another must be 0 somewhere on the interval.

A continuous function on a closed bounded interval is bounded and attains its bounds.