Could you explanin how to prove 2.31 using definitions below? Theorem 2.30. Let A be a subset of the topological space X, and let p be a point in X. Ifthe set {x;}ien CA and x; → p, then p is in the closure of A.As we shall see later, in some topological spaces, the converse of the previous resultis not true. But it is true for R".Theorem 2.31. In the standard topology on R", if p is a limit point of a set A, then thereis a sequence of points in A that converges to p.Definition. Let (X,T) be a topological space, A a subset of X, and pa point in X. Then pis a limit point of A if and only if for each open set U containing p, (U – {p}) N A # Ø.Notice that p may or may not belong to A.Definition. A sequence in a topological space X is a function from N toX. The image ofi under this function is a point of X denoted x; and we traditionally write the sequenceby listing its images: x1, X2, X3, ... or in shorter form: (x¡)ieN-Definition. A point p E X is a limit of the sequence (x;)ieN, or, equivalently, (x;)ieNconverges to p (written x; → p), if and only if for every open set U containing p, thereis an NEN such that for all i > N, the point x¡ is in U.

now suppose that p is a limit point of A .then for each n∈N we have 1n>0,hence there exists some…

Answered: Theorem 2.31. In the standard topology…

Theorem 2.31. In the standard topology on R", if p is a limit point of a set A, then there is a sequence of points in A that converges to p.

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter1: Fundamentals

Section1.7: Relations

Problem 23E

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Concept explainers

Correlation

Correlation defines a relationship between two independent variables. It tells the degree to which variables move in relation to each other. When two sets of data are related to each other, there is a correlation between them.

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A correlation is used to determine the relationships between numerical and categorical variables. In other words, it is an indicator of how things are connected to one another. The correlation analysis is the study of how variables are related.

Regression Analysis

Regression analysis is a statistical method in which it estimates the relationship between a dependent variable and one or more independent variable. In simple terms dependent variable is called as outcome variable and independent variable is called as predictors. Regression analysis is one of the methods to find the trends in data. The independent variable used in Regression analysis is named Predictor variable. It offers data of an associated dependent variable regarding a particular outcome.

Question

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Could you explanin how to prove 2.31 using definitions below?

Theorem 2.30. Let A be a subset of the topological space X, and let p be a point in X. If
the set {x;}ien CA and x; → p, then p is in the closure of A.
As we shall see later, in some topological spaces, the converse of the previous result
is not true. But it is true for R".
Theorem 2.31. In the standard topology on R", if p is a limit point of a set A, then there
is a sequence of points in A that converges to p.
Definition. Let (X,T) be a topological space, A a subset of X, and pa point in X. Then p
is a limit point of A if and only if for each open set U containing p, (U – {p}) N A # Ø.
Notice that p may or may not belong to A.
Definition. A sequence in a topological space X is a function from N toX. The image of
i under this function is a point of X denoted x; and we traditionally write the sequence
by listing its images: x1, X2, X3, ... or in shorter form: (x¡)ieN-
Definition. A point p E X is a limit of the sequence (x;)ieN, or, equivalently, (x;)ieN
converges to p (written x; → p), if and only if for every open set U containing p, there
is an NEN such that for all i > N, the point x¡ is in U.

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