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Thread: Abstract Inequalities

  1. #1
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    Abstract Inequalities

    Can anyone help me confirm if I've solved this correctly?

    Many thanks.

    Q.
    If $\displaystyle a^2+b^2=1$ & $\displaystyle c^2+d^2=1$, show that ac + bd < 1.

    Attempt: $\displaystyle a^2+b^2+c^2+d^2=2$
    if $\displaystyle a^2+c^2\geq2ac$ & $\displaystyle b^2+d^2\geq2bd$
    if $\displaystyle 2ac+2bd\leq2$
    if $\displaystyle ac+bd\leq1$...true
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  2. #2
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    Re: Abstract Inequalities

    Yes, that is a perfectly good proof.

    (Of course, $\displaystyle a^2+ c^2\ge 2ac$ because $\displaystyle (a- c)^2= a^2- 2ac+ c^2\ge 0$.)
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  3. #3
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    Re: Abstract Inequalities

    Great. Thank you.
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