Answer to Question #288896 in Real Analysis for Dhruv bartwal

Question

Give an example of an infinite set with finite number of limit points, with proper justification.

Answer 1

$\text{The set}\\ \Big\{\frac{1}{n} \mid n \in \mathbb{Z}^+\Big\} \bigcup \\ \Big\{1+ \frac{1}{n} \mid n \in \mathbb{Z}^+\Big\} \\ \text{ is an infinite set which has a finite limit point} \\ \text{By the definition of limit, we can write}\\ \Big\{\frac{1}{n} \mid n \in \mathbb{Z}^+\Big\} \\ \text{as}\\ \lim_{n \to \infty} \frac{1}{n} = 0 \\ \text{while for}\\ \Big\{1+ \frac{1}{n} \mid n \in \mathbb{Z}^+\Big\}\\ \text{write as}\\ \lim_{n \to \infty} (1 + \frac{1}{n}) = 1 \\ \text{Hence,}\\ \Big\{\frac{1}{n} \mid n \in \mathbb{Z}^+\Big\} \bigcup \\ \Big\{1+ \frac{1}{n} \mid n \in \mathbb{Z}^+\Big\} = \{0,1\} \, \text{which is finite.}$