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Math Help - New to proofs...

  1. #1
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    New to proofs...

    If a \leq b and b \leq a+c, then  |a-b| \leq c.

    Ok this is what I need to prove. I am fairly new to this so i'm not too sure what i have done is right, but here is what i have.

    a \leq b \leq a+c (subtract a)

    0 \leq b-a \leq c (multiply by -1)

    0 \geq a-b \geq -c so (??? not sure after this point ???) c \geq 0. therefor c \geq 0 \geq a-b \geq -c so c \geq a-b \geq -c which equals |a-b| \leq c.

    Please if i have don something wrong correct me or if there is a better way of going about this that would be helpful too. Thanks for you help.
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  2. #2
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    You have gotten to the point \displaystyle 0 \leq b-a \leq c.

    Since \displaystyle b-a is a positive value, it's equal to its absolute value.

    So \displaystyle |b - a| \leq c.


    But \displaystyle |b-a| = |a -b|.

    Therefore \displaystyle |a - b| \leq c.

    Q.E.D.
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