I'm trying to prove that $\pi_1(S^n)=0$ when $n>1$.
I've noticed that if you just take the north pole and south pole from each sphere, you can apply Van Kampen theorem. This doesn't fail as when $n>1$ it's still simply connected as it's homotopic to a circle.
But, how would you make this argument more formal? Would you use induction. But, can't see what the induction step will be. Help?