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.

Hint: Recall the Cauchy-Schwarz Inequality.
$$|\langle x,y\rangle| ^2 \leq \langle x,x\rangle \cdot \langle y,y\rangle,$$
$$|\langle x,y\rangle| \leq \|x\| \cdot \|y\|.$$ 
You can explore its various applications and extensions at the Wikipedia Entry linked above.
