Accumulation points of sets

Statement

Indicate all the accumulation points of:

• (a) $A = [0,2)$;
• (b) $B = \{(-1)^n + 1/n : n\ge 1\}$;
• (c) $C = \mathbb{Q}\cap[0,1]$;
• (d) $D = \{n^2 : n\in\mathbb{N}\}$.

Solution

• (a) The interval $[0,2)$ has as its accumulation points the whole closed interval: the set is $[0,2]$. In particular $2$ is an accumulation point even though it does not belong to $A$.
• (b) We separate the even and odd terms. For $n$ even: $(-1)^n+1/n = 1+1/n \to 1$. For $n$ odd: $-1+1/n \to -1$. The accumulation points are $\boxed{\{-1,\ 1\}}$.
• (c) The rationals of $[0,1]$ are dense in $[0,1]$: every real of $[0,1]$ is an accumulation point. The set of accumulation points is $[0,1]$.
• (d) The squares $0,1,4,9,16,\ldots$ get further and further apart: every point is isolated and there is no finite accumulation point (only $+\infty$).

Links

Topics: Limits
Concepts: Accumulation point
Skills: Reasoning by cases
Exercise type: True or false