The distance function on the real line
The absolute value |x| represents the distance of x from zero on the number line. It is always non-negative.
Non-negativity: absolute value is always non-negative
Zero property: only zero has absolute value zero
Symmetry: opposite numbers have equal absolute values
Multiplicativity: absolute value of product is product of absolute values
Divisibility: absolute value of quotient (y ≠ 0)
Triangle inequality: fundamental for analysis
Reverse triangle inequality
Distance symmetry: distance is symmetric
Square relation: useful for algebraic manipulations
Bounds: x is bounded by its absolute value
The triangle inequality |x + y| ≤ |x| + |y| is one of the most important inequalities in analysis. It's used in proofs involving limits, continuity, and metric spaces. The name comes from the fact that in any triangle, the sum of two sides is at least as long as the third side.
A set S ⊆ ℝ is bounded above if there exists M ∈ ℝ such that x ≤ M for all x ∈ S.
Example:
S = (-∞, 5) is bounded above by 5 (or any M ≥ 5)
A set S ⊆ ℝ is bounded below if there exists m ∈ ℝ such that x ≥ m for all x ∈ S.
Example:
S = [3, ∞) is bounded below by 3 (or any m ≤ 3)
A set is bounded if it is both bounded above and bounded below. Equivalently:
| Set | Bounded? | Sup | Inf | Notes |
|---|---|---|---|---|
| [0, 1] | Yes | 1 (attained) | 0 (attained) | Closed bounded interval |
| (0, 1) | Yes | 1 (not attained) | 0 (not attained) | Open bounded interval |
| {1/n : n ∈ ℕ} | Yes | 1 (attained) | 0 (not attained) | Sequence tending to 0 |
| [0, ∞) | Below only | ∞ | 0 (attained) | Unbounded above |
| ℤ | No | ∞ | -∞ | Unbounded in both directions |
| {(-1)ⁿ/n : n ∈ ℕ} | Yes | 1/2 (attained) | -1 (attained) | Alternating sequence |
| {r ∈ ℚ : r² < 2} | Yes | √2 (not in set) | -√2 (not in set) | ℚ lacks sup (√2 ∉ ℚ) |
The least upper bound and greatest lower bound
means:
Equivalent:
means:
Equivalent:
Every non-empty subset of ℝ that is bounded above has a supremum in ℝ. Every non-empty subset of ℝ that is bounded below has an infimum in ℝ.
The completeness principle distinguishes ℝ from ℚ. It guarantees that limits exist, that continuous functions on closed intervals attain their bounds, and is the foundation of calculus. Without it, we couldn't define integrals or prove the Intermediate Value Theorem.
Five equivalent ways to express the completeness of ℝ
Every non-empty set bounded above has a supremum
Standard definition of completeness
Every bounded monotonic sequence converges
Essential for series convergence proofs
Nested closed intervals with lengths → 0 have exactly one common point
Bisection method convergence
Every bounded sequence has a convergent subsequence
Compactness arguments
Every Cauchy sequence converges
Metric space completeness
Closed bounded sets are compact
Covering arguments in analysis
For non-negative reals a₁, a₂, ..., aₙ:
Equality Condition:
Equality holds if and only if a₁ = a₂ = ⋯ = aₙ
Example:
For a = 3, b = 12: (3+12)/2 = 7.5 ≥ √36 = 6 ✓
For x ≥ -1 and n ∈ ℕ:
Equality Condition:
Equality holds if and only if x = 0 or n = 1
Example:
For x = 0.5, n = 3: 1.5³ = 3.375 ≥ 1 + 1.5 = 2.5 ✓
For reals a₁, ..., aₙ and b₁, ..., bₙ:
Equality Condition:
Equality holds if aᵢ = k·bᵢ for some constant k
Example:
For (1,2) and (3,4): (1·3+2·4)² = 121 ≤ (1+4)(9+16) = 125 ✓
For positive reals, as r increases, the r-power mean increases:
Equality Condition:
All means are equal if and only if all values are equal
Example:
HM ≤ GM ≤ AM generalizes to any power means
Problem:
Solution:
First, list some elements: S = {1/2, 2/3, 3/4, 4/5, ...}
As n → ∞, n/(n+1) → 1, so 1 is an upper bound
For any ε > 0, choose n > 1/ε - 1, then n/(n+1) > 1 - ε
So 1 is the least upper bound: sup S = 1
Note: 1 ∉ S, so max S does not exist
The smallest element is 1/2 when n = 1
So inf S = min S = 1/2
Problem:
Prove that for a, b ≥ 0: (a+b)/2 ≥ √(ab)
Solution:
Start with (√a - √b)² ≥ 0 (always true for reals)
Expand: a - 2√(ab) + b ≥ 0
Rearrange: a + b ≥ 2√(ab)
Divide by 2: (a+b)/2 ≥ √(ab) ∎
Equality holds when √a = √b, i.e., a = b
Problem:
Prove (1+x)ⁿ ≥ 1 + nx for x ≥ -1, n ∈ ℕ
Solution:
Base case (n = 1): (1+x)¹ = 1 + x ≥ 1 + 1·x ✓
Inductive step: Assume (1+x)ᵏ ≥ 1 + kx for some k
(1+x)ᵏ⁺¹ = (1+x)ᵏ · (1+x)
≥ (1 + kx)(1 + x) by inductive hypothesis (note 1+x ≥ 0)
= 1 + x + kx + kx²
= 1 + (k+1)x + kx²
≥ 1 + (k+1)x since kx² ≥ 0 ∎
Problem:
Prove that ||a| - |b|| ≤ |a - b|
Solution:
From triangle inequality: |a| = |(a-b) + b| ≤ |a-b| + |b|
So |a| - |b| ≤ |a-b|
Similarly: |b| = |(b-a) + a| ≤ |b-a| + |a| = |a-b| + |a|
So |b| - |a| ≤ |a-b|, i.e., -(|a| - |b|) ≤ |a-b|
Combining: -|a-b| ≤ |a| - |b| ≤ |a-b|
Therefore: ||a| - |b|| ≤ |a-b| ∎
Problem:
Find the minimum value of x + 4/x for x > 0
Solution:
By AM-GM inequality: (a + b)/2 ≥ √(ab) for positive a, b
Let a = x and b = 4/x
Then (x + 4/x)/2 ≥ √(x · 4/x) = √4 = 2
So x + 4/x ≥ 4
Equality holds when x = 4/x, i.e., x² = 4, x = 2 (since x > 0)
Minimum value is 4, achieved at x = 2 ∎
Problem:
Prove that (∑aᵢbᵢ)² ≤ (∑aᵢ²)(∑bᵢ²) for real numbers
Solution:
Consider f(t) = ∑(aᵢ + tbᵢ)² ≥ 0 for all t ∈ ℝ
Expand: f(t) = ∑aᵢ² + 2t∑aᵢbᵢ + t²∑bᵢ²
Let A = ∑aᵢ², B = ∑aᵢbᵢ, C = ∑bᵢ²
So f(t) = Ct² + 2Bt + A ≥ 0
A non-negative quadratic has discriminant ≤ 0
(2B)² - 4AC ≤ 0
4B² ≤ 4AC, so B² ≤ AC
Therefore (∑aᵢbᵢ)² ≤ (∑aᵢ²)(∑bᵢ²) ∎
Problem:
Find inf S and determine if it's a minimum, where S = {1/n : n ∈ ℕ, n ≥ 1}
Solution:
Elements of S: 1, 1/2, 1/3, 1/4, ...
All elements are positive, so 0 is a lower bound
For any ε > 0, by Archimedean property ∃n with 1/n < ε
So 0 is the greatest lower bound: inf S = 0
But 0 ∉ S (no n gives 1/n = 0)
Therefore inf S = 0, but min S does not exist ∎
Problem:
Find the maximum of x²y³ subject to 2x + 3y = 10, x,y > 0
Solution:
By AM-GM: (a₁ + a₂ + ... + aₙ)/n ≥ (a₁a₂...aₙ)^(1/n)
Write 2x + 3y = x + x + y + y + y = 10
Apply AM-GM to these 5 terms: (x + x + y + y + y)/5 ≥ (x·x·y·y·y)^(1/5)
So 2 ≥ (x²y³)^(1/5), meaning x²y³ ≤ 32
Equality when x = y, from 2x + 3y = 10 we get 5x = 10, x = 2
Maximum is 32, achieved at x = y = 2 ∎
If α = sup S and ε > 0, then ∃x ∈ S: α - ε < x ≤ α
Application: Used to show sequences converging to supremum exist
Every bounded monotonic sequence converges
Application: Fundamental for series convergence and integration
Every bounded sequence has a convergent subsequence
Application: Essential for compactness arguments and analysis
For positive reals and weights summing to 1: weighted AM ≥ weighted GM
Application: Extends to weighted averages and probability
For r < s: M_r ≤ M_s where M_r = (∑xᵢʳ/n)^(1/r)
Application: Generalizes AM-GM to continuous scale of means
For a,b ≥ 0 and p,q > 1 with 1/p + 1/q = 1: ab ≤ aᵖ/p + bᵠ/q
Application: Used in proving Hölder's inequality
|∑xᵢ| ≤ ∑|xᵢ| for any finite collection
Application: Error bounds in numerical computations
To prove α = sup S, show: (1) α is an upper bound, (2) for any ε > 0, ∃x ∈ S with x > α - ε
Example: Proving sup(0,1) = 1: 1 bounds (0,1), and 1-ε/2 ∈ (0,1) for any ε > 0
Many inequalities become easier after squaring both sides (when both sides are non-negative)
Example: To prove AM ≥ GM, start with (√a - √b)² ≥ 0
Prove inequality for n=1, then show if true for n, it holds for n+1
Example: Bernoulli's inequality (1+x)ⁿ ≥ 1+nx is proven by induction on n
To disprove 'x is infinitely small', find n with 1/n < x
Example: There is no smallest positive real: for any x > 0, x/2 is smaller and positive
Rewrite products as telescoping to simplify bounds
Example: ∏(1+1/k²) telescopes nicely when written as ∏((k²+1)/k²)
For convex f: f(∑λᵢxᵢ) ≤ ∑λᵢf(xᵢ) when ∑λᵢ = 1
Example: Proving AM-GM using convexity of -ln(x)
Equality in AM-GM achieved when all variables equal
Example: min(x+y+z) s.t. xyz=1: use x=y=z=1, minimum is 3
Developed rigorous definitions for limits and continuity, introduced the ε-δ formalism
His work on limits laid the foundation for modern analysis and the formal treatment of inequalities
Perfected Cauchy's definitions and proved the Bolzano-Weierstrass theorem
Established the importance of supremum and infimum in analysis, formalized the concept of bounded sets
Early work on the least upper bound property and intermediate value theorem
Recognized the need for a rigorous foundation of real numbers before Dedekind and Weierstrass
AM-GM inequality is used to find optimal dimensions in engineering and economics
Example: Minimizing surface area of a box with fixed volume: make it a cube
Cauchy-Schwarz inequality bounds correlations and covariances
Example: Proving that correlation coefficient |ρ| ≤ 1
Triangle inequality describes the metric structure of space
Example: Proving the uncertainty principle in quantum mechanics
Bounds on approximations and error analysis
Example: Analyzing floating-point arithmetic errors using absolute value inequalities
The supremum (least upper bound) always exists for non-empty bounded-above sets of reals, but may not be an element of the set. The maximum is the largest element of the set and must belong to the set. If the maximum exists, it equals the supremum. Example: S = (0,1) has sup S = 1 but no maximum.
In ℚ, the set {x ∈ ℚ : x² < 2} is bounded above but has no supremum in ℚ (would need √2). The completeness axiom guarantees that every non-empty bounded-above set of reals HAS a supremum, which is essential for limits, continuity, and integration.
A set S is bounded above if ∃M: ∀x ∈ S, x ≤ M. Bounded below if ∃m: ∀x ∈ S, x ≥ m. Bounded means both. To show unboundedness, prove that for any M, you can find x ∈ S with x > M (or x < M for below).
Equality in AM-GM holds if and only if all numbers are equal: a₁ = a₂ = ⋯ = aₙ. This is crucial for optimization: to minimize a sum given a fixed product (or vice versa), make all terms equal.
Cauchy-Schwarz is extremely versatile. It's used to prove other inequalities, bound dot products, in optimization problems, and even in probability (correlation bounds). The key insight: it measures how 'aligned' two vectors are.
If α = sup S, then for any ε > 0, there exists x ∈ S such that α - ε < x ≤ α. This says we can always find elements of S arbitrarily close to the supremum. This property is often used in proofs involving limits.
The Archimedean property (for any x, ∃n ∈ ℕ with n > x) follows from the completeness axiom. It's equivalent to saying ℕ is unbounded above, which prevents 'infinitely large' or 'infinitely small' real numbers from existing.
AM-GM is used in optimization (minimizing sums with fixed products). Cauchy-Schwarz appears in statistics (covariance bounds), physics (uncertainty principles), and machine learning (cosine similarity). Bernoulli's inequality is key for proving limits involving powers.
Test your understanding of real number properties