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Math Help - Sequences and proofs

  1. #1
    Junior Member
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    Sequences and proofs

    I have a sequence {a[n]} from n=1 to infinity, a is a real number

    Show that if |a[n] -a|<1

    then

    i) |a[n]|</= |a| + 1
    ii) |a[n] + a|</= 2|a|+1
    iii) |a[n]^2 - a^2| </= |a[n] -a|(2|a|+1)

    How are these done? I'm familiar with the triangle inequality and such, but these don't seem familiar. Perhaps I'm missing something basic or elementary? Any comments welcome, cheers
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  2. #2
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    This is a basic fact: \left| a \right| - \left| b \right| \leqslant \left| {\left| a \right| - \left| b \right|} \right| \leqslant \left| {a - b} \right|.

    From that you can prove many things.
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