Let's think geometrically.

By definition

Let us draw a square of side-length first, then put another one but with side-length beisde the other one. So that they share a side. Now we can put a square of side right next to the other ones so that a border is shared and so on...

For example see here

Note now that, the area of the big rectangle is equal to the sum of the areas of the small squares inside. So, if the sides of the rectangle are and , it follows that

You may also try using induction