Stuck on Schwartz Inequality (Equality)

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I am stuck on problem 18.(b) of chapter one of Calculus by Michael Spivak. I cannot seem to prove equality for the first case it poses, I seem to only be able to prove the inequality. Can someone please give me a hint as to what I'm doing wrong? I don't want the full answer, if I did I would just consult the answer manual, which I have. My working is attached.