I am to prove CS inequality by mathematical induction. Don't even know where to start. Any help?
Perhaps it's Cauchy-Schwartz?
The proof I usually use for Cauchy-Schwartz, in the form $\displaystyle \Vert u \Vert^2 \Vert v \Vert^2 \ge (u\cdot v)^2$, is to consider the quadratic $\displaystyle f(x) = \Vert u + xv \Vert^2 \ge 0$ and observe that the requirement ("b^2-4ac") that f has at most one real root is precisely C-S. This doesn't depend on dimension.
If I had to do it by induction I think I'ld start by observing that it is trivial for n=1. The case n=2 follows from $\displaystyle (x^2+y^2)(u^2+v^2) = (xu+yv)^2 + (xv-yu)^2 \ge (xu+yv)^2$. The induction step would be something like $\displaystyle ((u_1^2+\cdots+u_{n-1}^2)+u_n^2)((v_1+\cdots+v_{n-1}^2)+v_n^2) \ge $ $\displaystyle \big\vert\sqrt{u_1^2+\cdots+u_{n-1}^2}\sqrt{v_1+\cdots+v_{n-1}^2}+u_nv_n \big\vert \ge $ $\displaystyle \big\vert(u_1,\ldots,n_{n-1})\cdot(v_1,\ldots,v_{n-1})+u_nv_n\big\vert $, thus using the cases 2 and n-1.