Accumulation point and limit point

In summary, the statement "If a sequence converges to L, then L is an accumulation point of {a_n|n greater than or equal to 1)" is not always true. A counterexample can be given by considering the sequence {a_n} where a_n=L for all n, which converges to L but has a finite range and therefore no accumulation point. In a Hausdorff space, an equivalent formulation of an accumulation point is "L is an accumulation point of S if every open set containing L contains a point in S other than L." This formulation shows that the counterexample given does not have an accumulation point.
  • #1
thesleeper
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Homework Statement


"If a sequence converges to L, then L is an accumulation point of {a_n|n greater than or equal to 1)."?
Prove or disprove the statement

Homework Equations


accumulation point is also a limit point


The Attempt at a Solution


I think the statement is not true. So in order to disprove it, I give an counterexample
consider the sequence {a_n} where a_n=L for all n. This sequence converges to 1, but its range is finite. Hence, this sequence has no accumulation point. Since the definition of an accumulation point of S is that every neighborhood of it contains infinitely many points of the S. Then, the statement is not always true.
Am I correct? and if the statement is true, how do you prove it?
Please help me with that. Thanks in advance
 
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  • #2
Welcome to PF!

It looks like you're on the right to me, though the question is pretty lousy. What have you got to doubt about your argument?

If it helps an equivalent formulation in a Hausdorff space (so in the real/complex numbers) is "L in a space X is an accumulation point of S iff every open set containing L contains a point in S other than L". What does this say about your example?
 

1. What is an accumulation point?

An accumulation point, also known as a cluster point or a limit point, is a point in a set of numbers where an infinite number of points in the set are arbitrarily close to it. In other words, any neighborhood of an accumulation point contains an infinite number of points from the set.

2. What is the difference between an accumulation point and a limit point?

While both terms are often used interchangeably, there is a subtle difference between them. An accumulation point is defined as a point where an infinite number of points in a set are arbitrarily close, while a limit point is defined as a point where every neighborhood contains at least one point from the set (excluding the point itself).

3. How are accumulation points and limit points used in calculus?

In calculus, accumulation points and limit points are used to define the concepts of limit and continuity. The limit of a function at a point is defined as the accumulation point of the function's values as the input approaches that point. Similarly, continuity of a function at a point is defined in terms of limit points.

4. Can a set have more than one accumulation point?

Yes, a set can have multiple accumulation points. In fact, some sets may have an infinite number of accumulation points, while others may have none at all.

5. How are accumulation points and limit points related to the concept of convergence?

Accumulation points and limit points are closely related to the concept of convergence. A sequence of numbers is said to converge to a limit point if the terms of the sequence get arbitrarily close to the limit point as the index goes to infinity. Similarly, a sequence of sets is said to converge to an accumulation point if the sets get arbitrarily close to the accumulation point.

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