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Advanced Calculus definitions for Test 2.

### 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|<∈

### frequently

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

### 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| <∈

### 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.

### 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.