Question

Construct a set of real numbers having exactly three limit points.

Answer 1

Think about the following set:
\[A:=\left\{x\in\mathbb{R}~:~x=\frac{1}{n},~n\in\mathbb{N}\right\}\]
How many limit points does it have?

Answer 2

The answer's not three, but how can you modify this set so that it does?

Answer 3

The problem is, I can't understand the idea of a limit point :( but as I googled A has 1 limit point right?

Answer 4

Correct.

I'm actually in the process of covering the same topic in real analysis. This video really helped me grasp the meaning of a limit point: http://www.youtube.com/watch?v=Ebnoxgp8mLM

Answer 5

You're studying this subject?

Answer 6

Yep

Answer 7

wow that cool, what's your major?

Answer 8

btw I should watch the full video ? 1h 12min?

Answer 9

Math.

Regarding that video, the instructor gets into the definition and a few examples around 12-20 min.

Answer 10

okay brb

Answer 11

why not 1 is the limit point?

Answer 12

for the 1/n set

Answer 13

Let's consider the definition of a limit point: For 1 to be a limit point of \(A\) (or \(G\), if you're still using the video), it has to satisfy the condition that *every* neighborhood of 1 contains a point in \(A\) (and this point can't be 1).

Consider the neighborhood \(N_{1/4}(1)\) (the open interval centered at 1 with radius 1/4):

Are there any points of \(A\) in this interval, other than 1?