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Math Help - use sequences to prove limit of |f(x)| HELP!

  1. #1
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    use sequences to prove limit of |f(x)| HELP!

    Suppose f -> R and define the function |f|: D -> R by |f|(x) = |f(x)|. Prove that if lim as x -> xo f(x) = L then lim as x->xo |f|(x) exists and equals |L|.

    I can do this using epsilons and deltas yet I am unsure as to how I would prove this using SEQUENCES.
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  2. #2
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    Quote Originally Posted by really.smarty View Post
    Suppose f -> R and define the function |f|: D -> R by |f|(x) = |f(x)|. Prove that if lim as x -> xo f(x) = L then lim as x->xo |f|(x) exists and equals |L|.

    I can do this using epsilons and deltas yet I am unsure as to how I would prove this using SEQUENCES.
    Let x_n be any sequence converging to x_0. The fact that \lim_{x\to x_0} f(x)= L tells you that \lim_{n\to \infty} f(x_n)= L. What can you say, then, about \lim_{n\to \infty} |f(x_n)|?
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  3. #3
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    it would equal |L| but i am unsure as to how this will help me answer the question?
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  4. #4
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    Hint, use the reverse triangle inequality, i.e. | |x| - |y| | \leq |x-y|
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