Understand the rigorous construction of real numbers and their fundamental properties
Real analysis begins with a rigorous foundation: understanding exactly what real numbers are and why they behave the way they do. The real numbers have a special property called completeness that distinguishes them from the rationals.
In this section, we'll explore the construction of real numbers and discover why completeness is essential for calculus to work.
One way to rigorously construct the real numbers is through Dedekind cuts: partitions of the rationals that define the "gaps" where irrational numbers live.
Key Insight
For √2, the set A has no maximum element—rationals get arbitrarily close to √2 from below but never reach it.
A Dedekind cut partitions the rationals Q into two sets (A, B) where:
Each cut defines a unique real number—the "gap" between A and B. This construction proves that R is complete: every bounded set has a least upper bound.
The defining property of real numbers: every non-empty set bounded above has a least upper bound (supremum). This property fails for the rationals!
Least Upper Bound Property: Every non-empty subset of R that is bounded above has a least upper bound (supremum) in R.
This property fails for Q! The set {q ∈ Q : q² < 2} is bounded above in Q but has no least upper bound in Q—the supremum would be √2, which is irrational.
For every real number x, there exists a natural number N greater than x. The natural numbers are unbounded above.
For x = 3.7000, the smallest natural number N > x is:
N = 4
Best approximation with denominator ≤ 100:
37/10 = 3.700000
Error: 0.000e+0
Statement: For every real number x, there exists a natural number N such that N > x.
Equivalently: the natural numbers are unbounded above. There is no real number larger than all natural numbers.
Consequence: For any ε > 0, there exists N such that 1/N < ε. This is why sequences like 1/n converge to 0.
Between any two distinct real numbers, there exists both a rational and an irrational number. Both sets are dense in R.
Rational found:
29/20 = 1.45000000
Irrational found:
1.4000 + √2×10⁻2 ≈ 1.41414214
Density of Rationals: Between any two distinct real numbers a < b, there exists a rational number r with a < r < b.
Density of Irrationals: Between any two distinct real numbers a < b, there exists an irrational number s with a < s < b.
No matter how small the interval, it always contains both rationals and irrationals! Try making the interval very small—you'll always find both types of numbers inside.
A sequence of nested closed intervals with lengths approaching zero contains exactly one point. This property is equivalent to completeness.
The intersection of all nested closed intervals is exactly one point:
⋂ Iₙ = {√2}
Theorem: If I₁ ⊇ I₂ ⊇ I₃ ⊇ ... is a nested sequence of non-empty closed bounded intervals with lengths → 0, then the intersection contains exactly one point.
This property is equivalent to completeness and is used to prove many fundamental results in analysis, including the existence of limits and the Bolzano-Weierstrass theorem.
Bisection: If m² < 2, take [m, b]; else take [a, m]. Limit = √2
Every non-empty subset of R that is bounded above has a least upper bound in R. This is the defining axiom of real numbers.
For every x ∈ R, there exists n ∈ N with n > x. Equivalently, for ε > 0, there exists N with 1/N < ε.
Both Q and R∖Q are dense in R: between any two reals lies both a rational and an irrational number.
If I₁ ⊇ I₂ ⊇ ... are closed bounded intervals with |Iₙ| → 0, then ⋂Iₙ contains exactly one point.