Chapter 7: Sequences and Convergence

Question 1

sequence

Answer 1

In a metric space (X, d) is a function x mapping the naturals to X.

Question 2

bounded

Answer 2

A sequence in metric space (X,d) is bounded if there exists a point p in X and B in R such that d(p, xn) less than or equal to B for all n in Naturals.

Question 3

converge

Answer 3

A sequence in a metric space (X,d) is said to converge to a point p in X if for every epsilon >0 there exists an M in Naturals such that d(xn, p) < epsilon for all n greater than or equal to M. the point p is said to be the limit of the sequence.

Question 4

Facts about sequences in Metric Spaces

Answer 4

1. A convergent sequence in a metric space has a unique limit
2. a convergent sequence in a metric space is bounded
3. a sequence {xn} in a metric space (X,d) converges to p in X if and only if there exists a sequence {an} of real numbers such that d(xn, p) is less than or equal to an for all n in the naturals, and the lim an = 0.

Question 5

More facts about sequences in Metric Spaces

Answer 5

Let {xn} be a sequence in a metric space (X,d).

1. if {xn} converges to p in X, then every subsequence of {xn} converges to p
2. if for some K in Naturals the K tail converges to p in X, then xn converges to p.

Question 6

Facts about Convergence in Euclidean Space

Answer 6

Let {x^j} be a sequence in Rn, where we write xj = (xj1, xj2, ..xjn) in Rn. Then {x^j} converges if and only if {xk^j} converges for every k, in which case
lim x^j = (lim x1j, lim x2j…, lim xnj)

Question 7

Facts about Convergence and Topology

Answer 7

1. let (X,d) be a metric space and {xn} a sequence in X. Then {xn} converges to x in X if and only if for every open neighborhood U of x, there exists an M in Naturals such that for all n greater than or equal to M we have xn in U.
2. Let (X,d) be a metric space, E in X a closed set and {xn} a sequence in E that converges to some x in X. Then x is in E.
3. Let (X,d) be a metric space and A in X. If x is in the closure of A, then there exists a sequence {xn} of elements in A such that lim xn = x .