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Math Help - inequality and fractions

  1. #1
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    inequality and fractions

    I have been working on this for a little bit now, but I can't get it.
     \mbox{If } a \leq b + c \mbox{ and } a,b,c \geq 0, \mbox{ show } \frac{a}{a+1} \leq \frac{b}{b+1} + \frac{c}{c+1}. Its the very first question in a series of questions on metric spaces, so I have a feeling I am missing something obvious. Thank you.
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  2. #2
    Senior Member JaneBennet's Avatar
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    Dec 2007
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    \color{white}.\quad. a\ \leqslant\ b+c\ \leqslant\ abc+2bc+b+c

    \Rightarrow\ a-abc+1-bc\ \leqslant\ bc+b+c+1

    \Rightarrow\ (a+1)(1-bc)\ \leqslant\ (b+1)(c+1)

    \Rightarrow\ \frac{1-bc}{(b+1)(c+1)}\ \leqslant\ \frac1{a+1}

    \Rightarrow\ \frac1{b+1}+\frac1{c+1}-1\ \leqslant\ \frac1{a+1}

    \Rightarrow\ 1-\frac1{a+1}\ \leqslant\ 1-\frac1{b+1}\,+\,1-\frac1{c+1}

    \Rightarrow\ \frac{a}{a+1}\ \leqslant\ \frac{b}{b+1}+\frac{c}{c+1}
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