Linear Algebra Questions - Another approach to the proof of the Cauchy-Schwarz Inequality is

Another approach to the proof of the Cauchy-Schwarz Inequality is suggested by the
figure below, which shows that in R2 or R3, ||proj u(v) || <= ||v||. Show that this is
equivalent to the Cauchy- Schwarz Inequality. * The R2 and R3 represent all real
numbers in 2 and 3 dimension if that is unclear.

See problems in attached file