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 $1$.