Answer to Question #203187 in Real Analysis for Rajkumar

Question

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

justification.

Answer 1

The set

$$\Big\{\frac{1}{n}\big| n\in \mathbb{Z}^+\Big\} \cup\Big\{1+\frac{1}{n}\big| n\in \mathbb{Z}^+\Big\}$$

is an infinite set, which has a finite limit.

We can see this directly or we can use the assertion of finding limits in calculus.

For:

$$\Big\{\frac{1}{n}\big| n\in \mathbb{Z}^+\Big\}\\ \lim_{n \rightarrow \infty}{\frac{1}{n}} = 0$$

While for:

$$\Big\{1+\frac{1}{n}\big| n\in \mathbb{Z}^+\Big\}\\ \lim_{n \rightarrow \infty}{1+\frac{1}{n}} = 1$$

Thus:

$$\Big\{\frac{1}{n}\big| n\in \mathbb{Z}^+\Big\} \cup\Big\{1+\frac{1}{n}\big| n\in \mathbb{Z}^+\Big\}= \{0,1\}$$

Which is finite.