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Example: Examples of Accumulation Points

(related to Definition: Accumulation Point (Real Numbers))

The following are examples of subsets of real numbers with accumulation points:

• The real sequence $\left(\frac 1n\right)_{n > 0}$ has the accumulation point $0,$ since for any $\epsilon > 0$ there is an index $n\in\mathbb N$ such that $|1/n - 0| < \epsilon.$
• By definition of convergence, the limit of every convergent sequence is at the same time its accumulation point.
• If $[a,b]$ is a real interval, then any real sequences $(x_n)_{n\in\mathbb N}$ with $x_n\in[a,b]$ is bounded, and contains according to the theorem of Bolzano-Weierstrass a convergent subsequence. Thus, any such sequence $(x_n)_{n\in\mathbb N}$ has an accumulation point.
• All points of the real interval $[a,b]$ are its accumulation points.
• All rational points in the real interval $[a,b]\cap\mathbb Q$ are its accumulation points.

The following are examples of subsets of real numbers without accumulation points:

• The set of natural numbers $\{0,1,2,\ldots\}$ has no accumulation points.
• The set of integers $\{\ldots,-2,-1,0,1,2,\ldots\}$ has no accumulation points.

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References

Bibliography

1. Kane, Jonathan: "Writing Proofs in Analysis", Springer, 2016