Advanced Calculus definitions for Test 2.

### Sequence

a function whose domain is the natural numbers

### converges

a sequence (aⁿ)→a (a real number) if, for every positive number ∈, there exists an N∈of the natural numbers such that whenever n≥N it follows that |aⁿ-a|<∈

### ∈-neighborhood of a

Given a real number a∈R and a positive number ∈≥0, the set V₋(a)={x∈R: |x-a|<∈}

### Diverge

a sequence that does not converge

### eventually

a sequence (aⁿ) is _________ in a set A⊆R if there exists an N∈N such that aⁿ∈A ∀n≥N

### frequently

a sequence (aⁿ) is __________ in a set A⊆R if, for every N∈N, there exists an n≥N such that aⁿ∈A

### bounded

a sequence (xⁿ) is _______ if there exists a number M>0 such that |xⁿ|≤M for all n∈N.

### increasing

a sequence (aⁿ) is ________ if aⁿ< aⁿ⁺¹ for all n∈N

### decreasing

a sequence (aⁿ) is ________ if aⁿ>aⁿ⁺¹ for all n∈N

### subsequence

Let (aⁿ) be a sequence of real numbers, and let n₁<n₂<n₃<... be an increasing sequence of natural numbers. Then the sequence aⁿ₁,aⁿ₂,aⁿ₃,....is a _______

### Cauchy sequence

a sequence (aⁿ) is called a ________ if, for every ∈>0, there exists an N∈N such that whenever m,n≥N it follows that |aⁿ -aⁿⁿ| <∈

### open

A set O⊆R is ______ if for all points a∈O there exists and ∈-neighborhood V₃(a) ⊆ O.

### limit point

A point x is a _____________of a set A if every ∈-neighborhood V₃(x) of x intersects the set A in some point other than x.

### isolated point

A point a∈A is an ____________ of A if it is not a limit.

### closed

A set F⊆R is _______ if it contains its limit points

### closure

Given a set A⊆R, let L be the set of all limit points of A. The _________ of A is defined to be A = A ∪ L.

### compact

A set K⊆R is ________ if every sequence in K has a subsequence that converges to a limit that is also in K.

### bounded

A set A⊆R is __________ if there exists M > 0 such that |a| ≤ M for all a∈A.

Example: