I thought this question is quite easy.
My proof is that
In R, the subset is compact if and only if is closed and bounded.
K is bounded since d(1/2, r) < 1 for any r in K, and
it is closed since its only limit point is 0 and it is in K.
Therefore K is compact.

However, the solution given by the MIT Open Course is much more complex and
I can't understand what this is about.

Is my solution valid? Can you help me understand the MIT Open course solution? 
I can't understand the part beginning from "Because epsilon > 0, there exists an N"

Problem:

![enter image description here](https://i.stack.imgur.com/fRUmP.png)

![enter image description here](https://i.stack.imgur.com/uf2jV.png)

MIT Opencourse Solution: 
![enter image description here](https://i.stack.imgur.com/fRUmP.png)

![enter image description here](https://i.stack.imgur.com/uf2jV.png)