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Prove that a finite set has no accumulation points.
Solution
A number a is an accumuation point accumuation point accumuation point (or limit point limit
point limit point) of a set A if and only if for any ? > 0 the there exists the ? -neighborhood of a
contains at least one point of A distinct from a . In other words every neighborhood of a contains
points of A different from a . Keep in mind that an accumulation point of a set may or may not
belong to the set.
A neighborhood of a point (any open interval containing the point) that does not contain the
point is called a deleted neighborhood deleted neighborhood deleted neighborhood of the point.
Thus, the set (0, 2) ? { 1} is a deleted neighborhood of 1.
Finite sets have no accumulation points since around every real number (inside or outside the
set) you can find a deleted neighborhood that does not contain elements of the set.

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Prove that a finite set has no accumulation points.SolutionA n.pdf

  • 1. Prove that a finite set has no accumulation points. Solution A number a is an accumuation point accumuation point accumuation point (or limit point limit point limit point) of a set A if and only if for any ? > 0 the there exists the ? -neighborhood of a contains at least one point of A distinct from a . In other words every neighborhood of a contains points of A different from a . Keep in mind that an accumulation point of a set may or may not belong to the set. A neighborhood of a point (any open interval containing the point) that does not contain the point is called a deleted neighborhood deleted neighborhood deleted neighborhood of the point. Thus, the set (0, 2) ? { 1} is a deleted neighborhood of 1. Finite sets have no accumulation points since around every real number (inside or outside the set) you can find a deleted neighborhood that does not contain elements of the set.