I am a 16 year old High school student and I have just started using Gilbert Strang's Linear algebra book. I have reached the second problem set in the first chapter which is about dot products and angles between vectors and I am stuck on this particular question rather I don't understand. I have tried thinking about this for some time and I can't wrap my head around this proof. 

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> 24. One line proof of the $|u \cdot U| \le 1$ for unit vectors $u = (u\_1, u\_2)$ and $U = (U\_1, U\_2)$
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> $|u\cdot U| ≤ |u\_1|\cdot |U\_1| + |u\_2|\cdot |U\_2| ≤ \frac{u\_1^2 + U\_1^2}{2} + \frac{u\_2^2 + U\_2^2}{2} = 1$
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I understand $|u \cdot U| \le |u\_1| \cdot |U\_1| + |u\_2| \cdot |U\_2|$ but I dont understand the $\frac{u\_1^2 + U\_1^2}{2} + \frac{u\_2^2 + U\_2^2}{2} = 1$ part of the inequality.