How can I prove that $|\langle w,v\rangle| \leq 1$? How can I go about proving that if $\|w\| = \|v\| = 1$, then $|\langle w, v\rangle| \leq 1$?
I know that $\langle w,v\rangle = w_1v_1 + w_2v_2 + \cdots $ but I am not sure how to use that to prove this.

Remember the definition of cosine in an inner product linear space: if the angle between two vectors $\,u,v\,$ is $\,\theta\,$ , then
$$\cos\theta:=\frac{\langle\,u,\, v\,\rangle}{||u||\,||v||}$$
Now just remember that $\,|\cos\theta|\le 1\,$ ...(This follows from Cauchy-Schwarz).
If you defined the inner product by means of the above equation then trigonometry came first and you also know that $\,|\cos\theta|\le 1\,$, so either way...
