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Fundamental theorem of arithmetic
Every positive integer except 1 can be expressed uniquely as a product of primes

Bernoulli's inequality
(for all n in N)(For all a >1)[(1+a)^n>=1+na]

A set S in R is said to be bounded above if
(There exists k in R)(For all x in S)(x<=k)

Supremum
 Definition of upper bound and least upper bound.
 ie (There exists e>0)(There exists b in S)(b>ae)
We write a=sup S

infimum
 a is a lower bound for S; that is
 (for all x in S)(x>=a)
 a is the greatest lower bound for S; that is
 (for all e >0)(There exists b in S)(b<a+e)
 b=inf S

Let R be a relation from X to Y. The domain of R is the set
D(R)={x in X there exists y in Y  (x,y) in R}

The range of R is the set
Ran(R)={y in Y there exists x in X  (x,y) in R}

The inverse of R is the Relation R^1 from Y to X
R^1= {(y,x) in YxX  (x,y) in R}

A relation R on X is said to be reflexive if
(For all x in X) (x,x) in R

R is said to be symmetric if
(For all x in X)(For all y in Y){[(x,y) in R] > [(y,x) in R]}

R is said to be transitive if
(For all x,y,z in X){[((x,y) in R) and ((y,z) in R)] > [(x,z) in R]}

Equivalence relation
If it is symmetric, transitive and reflexive

Let R be an equivalence relation on X. Let x be in X. The equivalence class of x wrt R is the set
[x]R = {y in X  (y,x) in R}

F:X>Y and A in X. Image of A under f is the set
f(A) = {f(x)  x in A}

f:X>Y. Then f is called injective
(For all x1 in X)(For all x2 in X)[(x1 not equal to x2) > (f(x1) not equal to f(x2))]

f: X>Y, then f is called a surjection if
 (for all y in Y)(there exists x in X)[f(x) =y]
 This means Ran(f)=Y

f:X>Y is called a bijection if
It is both an injection and a surjection

The Archimedian Principle
Let x be in R. Then there exists n in Z such that x<n

A sequence converges to a limit a in R if
(For all e >0)(There exists N in N+)(for all n in N+)[(n>N)>(ana <e)

The sequence an diverges to infinity if
(For all M in R)(There exists N in N+)(For all n in N+)[(n>N)>(an<M)]

If a sequence of real numbers is bounded above and increasing then it is
convergent

If a sequence of real numbers is bounded below and decreasing then it is
convergent

If the series Sum (an) from n=1 to infinity is convergent then
lim n> infinity is 0

