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Math Help - convergent sequence

  1. #1
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    convergent sequence

    can someone please help me prove this statement,
    two sequences (an) and (bn) converge if and only if (an + bn) and (an - bn) converge.
    thank you.
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  2. #2
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    The whole turns on a simple application of the triangle inequality.
    \left| {\left( {a_n  + b_n } \right) - \left( {A + B} \right)} \right| \le \left| {a_n  - A} \right| + \left| {b_n  - B} \right|
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  3. #3
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    so the left hand side of the inequality you wrote is less than epsilon. how do we know that the two on the right are less than epsilon as well?
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  4. #4
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    Say \left( {a_n } \right) \to A\quad \& \quad \left( {b_n } \right) \to B.
    If \varepsilon  > 0 find an N such that n \ge N\quad  \Rightarrow \quad \left| {\left( {a_n } \right) - A} \right| < \frac{\varepsilon }{2}\quad \& \quad \left| {\left( {b_n } \right) - B} \right| < \frac{\varepsilon }{2}.

    You should understand how to do these if you are expected to complete these problems.
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  5. #5
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    ok so how do you know that abs(an - A) + abs(bn - B) is less than epsilon? you know what i'm saying? we know that that expression and that epsilon are both greater than or equal to abs((an + bn) - (A+B)). how do we know that abs(an - A) + abs(bn - B) is less than epsilon?

    you said that abs(an - A) is less than epsilon/2, but we don't know this, right?

    it's like the triangle inequality doesn't work because it's reversed.
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  6. #6
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    Please answer the following question in some detail
    Exactly what must one show in order to prove that <br />
\left( {a_n  + b_n } \right) \to A + B?
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  7. #7
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    somehow i think you're missing my question.

    i know that (an + bn) converges to A + B from the statement.

    i need to show that (an) converges to A and (bn) converges to B.

    are you we talking about the same thing?
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  8. #8
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    The question is an "if and only if question".
    That is you must do it both ways.
    It does not seem that you can do it either way.
    Have you done it the way that I started you off?
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  9. #9
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    Quote Originally Posted by CindyMichelle View Post
    somehow i think you're missing my question.
    i know that (an + bn) converges to A + B from the statement.
    i need to show that (an) converges to A and (bn) converges to B.
    No I did not miss your point. But rather tried to help see the error of your ways.
    Consider: \left( {a_n } \right) = \left( { - 1} \right)^n \quad \& \quad \left( {b_n } \right) = \left( { - 1} \right)^{n + 1} .

    Is it true that \left( {a_n  + b_n } \right) \to \left( {1 - 1} \right)?
    Is either of these true: \left( {a_n } \right) \to \left( 1 \right)\quad \mbox{or}\quad \left( {b_n } \right) \to \left( { - 1} \right).

    In fact is it possible for either \left( {a_n } \right)\quad \mbox{or}\quad \left( {b_n } \right) to converge.
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