How to prove by induction the constructibility of a line segment of length $\sqrt{n}$? How to prove the following statement by induction? 
If a line of unit length is given, then a line of length $\sqrt{n}$ can be constructed with straightedge and compass for each positive integer $n$. 
The statement obviously holds for $n=1$. 
Now we assume that we can construct a line of length $\sqrt{k}$ with straightedge and compass, where $k$ is a given positive integer. 
How to proceed from here? 
That is, how to show from this that a line of length $\sqrt{k+1}$ can also be constructed with straightedge and compass? 

Use Pythagoras' theorem for the right-angled triangle with legs $\sqrt{k}$ and $1$.
