Prove that $(ax + by + cz)^2 \leq (a^2 + b^2 + c^2)(x^2 + y^2 + z^2)$. Prove that for any real number $a$, $b$, $c$, $x$, $y$ and $z$, there results
$(ax + by + cz)^2 \leq (a^2 + b^2 + c^2)(x^2 + y^2 + z^2)$
I have thought this Q for long time but I still can't get the answer.
Can anyone help me please?
Thank you!~

We observe that
\begin{eqnarray*}
\text{RHS}&=&(a^2x^2+b^2y^2+c^2z^2)+(a^2y^2+b^2x^2)+(a^2z^2+c^2x^2)+(+b^2z^2+c^2y^2)\\
&=&(a^2x^2+b^2y^2+c^2z^2+2axby+2bycz+2czax)+(a^2y^2+b^2x^2-2aybx)+(a^2z^2+c^2x^2-2azcx)+(b^2z^2+c^2y^2-2bzcy)\\
&=&(ax+by+cz)^2+(ay-bx)^2+(az-cx)^2+(bz-cy)^2\\
&\geq&(ax+by+cz)^2\\
&\geq&\text{LHS}.
\end{eqnarray*}
