IP

Schröder–Bernstein Theorem

If f : X → Y and g : Y → X are injections, then there exists a bijection h : X → Y , and hence X and Y have the same cardinality

Least Upper Bound Axiom

Every non-empty subset of R that is bounded above has a supremum

Lagrange’s Theorem

Raising any element of a subgroup to the order of the subgroup results in the identity element of the group

Corollary to Lagrange’s Theorem

If G is a finite group, then the order of any element of G divides |G|

Sandwich Theorem

Suppose we have two convergent real sequences (an)n∈N and (cn)n∈N, and a real sequence (bn)n∈N, such that the following conditions hold:

(i) for all but finitely many values of n, an ≤ bn ≤ cn

(ii) as n→∞,an →a and cn →a.

Then bn →a as n→∞.

Hierarchy of limits (as n→ infinity)

n^n

n!

x^n (x>1)

n^q

Hierarchy of limits (n→0)

n^-q

x^n (|x|<1)

1/n!

n^-n

Principle of Bounded Monotone Convergence

If (an)n∈N is an increasing sequence which is bounded above then it converges to sup{an | n ∈ N} as n → ∞.

If (an)n∈N is a decreasing sequence which is bounded below then it converges to inf{an | n ∈ N} as n → ∞.

Bolzano-Weierstrass Theorem

Every real bounded sequence has a convergent subsequence

Cauchy Sequence

A real sequence (an)n∈N is called a Cauchy sequence if for any ε > 0 there exists an N ∈ N such that if m, n ≥ N then |am − an| < ε.

Limit Comparison Test

Suppose (aj)j∈N and (bj)j∈N are strictly positive sequences and that aj/bj converges to a finite limit L as j → ∞. Then:

(i) If P bj converges, then P aj converges.

(ii) If L > 0 then P aj and P bj either both converge or both diverge

Intermediate Value Theorem

Suppose f : X → R is continuous and a,b∈X with a<b. If d is any number such that f(a)≤d≤f(b) or f(b)≤d≤f(a) then there exists c ∈ [a,b] such that f(c) = d.