Math

Uniform Continuity

A function f is uniformly continuous on X if for every ε > 0 there is

δ = δ(ε) > 0 such that

x, y ∈ X, |x − y| < δ(ε) ⟹ |f(x) − f(y)| < ε.

Lipschitz

A function f: X → R is Lipschitz continuous on X if there exists a constant L > 0 such that for all x, y ∈ X,

|f(x) − f(y)| ≤ L|x − y|.

Lower Bound

Let X ⊂ R be nonempty. A number m, not necessarily in X, is a lower bound for X if x ≥ m for all x ∈ X.

If a set X has a lower bound, the set is called bounded from below.

Upper Bound

Let X ⊂ R be nonempty. A number M, not necessarily in X, is an upper bound for X if x ≤ M for all x ∈ X.

If a set X has an upper bound, the set is called bounded from above.

Bounded

A nonempty set X ⊂ R is said to be bounded if it is both bounded from below and bounded from above.

Triangle Inequality

For any x, y ∈ R:

|x + y| ≤ |x| + |y|

Reverse Triangle Inequality

For any x, y ∈ R:

|x − y| ≥ ||x| − |y||

Every covergent sequence is…

bounded

Squeeze Theorem for Sequences

Let {xₙ}ₙ₌₁∞, {yₙ}ₙ₌₁∞, and {zₙ}ₙ₌₁∞ be sequences in ℝ such that

for some n₀ ∈ ℕ⁺, xₙ ≤ yₙ ≤ zₙ for all n ≥ n₀,

and limₙ→∞ xₙ = limₙ→∞ zₙ = l.

Then {yₙ} converges and

limₙ→∞ yₙ = l.

monotone and bounded implies convergence

If {xₙ}ₙ₌₁∞ ⊂ ℝ is increasing and bounded above, then it converges and

limₙ→∞ xₙ = sup{xₙ : n ∈ ℕ⁺}.

If {xₙ}ₙ₌₁∞ ⊂ ℝ is decreasing and bounded below, then it converges and

limₙ→∞ xₙ = inf{xₙ : n ∈ ℕ⁺}.

Bolzano-Weierstrass

If {xₙ}ₙ₌₁∞ ⊂ ℝ is a bounded sequence, then it has a convergent subsequence.

Cauchy

A sequence {xₙ}ₙ₌₁∞ is Cauchy if and only if it is convergent; every Cauchy sequence is bounded

Composition of Limits

If lim(x→x₀) f(x) = y₀ and lim(y→y₀) g(y) = l, then lim(x→x₀) g(f(x)) = l

A function f : (a, b) → R is continuous at x₀ ∈ (a, b) if and only if…

lim(x → x₀) f(x) = f(x₀).

Archimedean Property

For every y > 0 and x ∈ R, there exists n ∈ N such that ny > x.

Limit

For every ε > 0, there exists δ > 0 such that whenever 0 < |x - x₀| < δ, then |f(x) - L| < ε.

Banach Fixed-Point Theorem

Let f : R → R be such that for some k < 1,

|f (x) − f (y)| ≤ k|x − y|, ∀x, y ∈ R.

There exists one and only one x ∈ R such that x = f (x).

Intermediate Value Theorem

Assume that f ∈ C⁰[a, b] is such that f(a) < f(b) (or f(a) > f(b)). For any y₀ ∈ R with f(a) < y₀ < f(b) (resp. f(a) > y₀ > f(b)), there exists x₀ ∈ (a, b) such that f(x₀) = y₀.