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Thread: Accumulation point

  1. #1
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    Accumulation point

    Prove that a real number s is an accumulation point of a set S if and only if there exists some sequence {a} in S such that a is not equal to s for every natural number n where the limit of a as n approaches infinity is equal to s.

    I'm not really sure how to start. I know since it's an if and only if statement we have to prove it both ways.
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  2. #2
    Senior Member Dinkydoe's Avatar
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    $\displaystyle s\in S$ is a accumulation point if for every $\displaystyle \epsilon > 0$ there exists a $\displaystyle x_0\in S$ with $\displaystyle a-\epsilon < x_0 < a+\epsilon$ and $\displaystyle x_0\neq a$

    Choose a $\displaystyle \epsilon > 0$.

    For every $\displaystyle n\geq 1$ you can choose a $\displaystyle x_n\in S$ with $\displaystyle x_n\neq a$ such that: $\displaystyle a-\frac{\epsilon}{n} < x_n < a+\frac{\epsilon}{n}$

    Now $\displaystyle x_n\to a$ as $\displaystyle n\to \infty$

    I'll leave the other implication to you.
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  3. #3
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    Quote Originally Posted by bethh View Post
    Prove that a real number s is an accumulation point of a set S if and only if there exists some sequence {a} in S such that a is not equal to s for every natural number n where the limit of a as n approaches infinity is equal to s.
    You did not tell us what sort of top-space we are considering.
    I will assume that you are working in a metric space (so the statement is true).
    If $\displaystyle r>0$ then $\displaystyle \mathcal{B}(a;r)=\{x:d(a,x)<r\}$ is a ball.
    Using the definition of accumulation point we get:
    $\displaystyle \left( {\exists x_1 } \right)\left[ {s \ne x_1 \,\& \,d\left( {x_1 ,s} \right) < 1} \right] $.
    $\displaystyle \left( {\exists x_2 } \right)\left[ {s \ne x_2 \,\& \,d\left( {x_2 ,s} \right) < \min(d\left( {x_1 ,s} \right),\frac{1}{2}} )\right]$.

    Now you have a way to generate your sequence by induction.
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  4. #4
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by bethh View Post
    Prove that a real number s is an accumulation point of a set S if and only if there exists some sequence {a} in S such that a is not equal to s for every natural number n where the limit of a as n approaches infinity is equal to s.

    I'm not really sure how to start. I know since it's an if and only if statement we have to prove it both ways.
    Let $\displaystyle \left\{x_n\right\}_{n\in\mathbb{N}}$ be a sequence in $\displaystyle S$ such that $\displaystyle x_n\ne x,\text{ }\forall n$ and $\displaystyle x_n\to x$. Now, let $\displaystyle B_{\varepsilon}(x)$ be arbitrary. Since $\displaystyle x_n\to x$ there exists an $\displaystyle N\in\mathbb{N}$ such that $\displaystyle N\leqslant n\implies x_n\in B_{\delta}(x)$ and since $\displaystyle x_N\ne x$ and $\displaystyle x_N\in B_{\delta}(x)$ we see there exists a point of $\displaystyle S$ distinct from $\displaystyle S$ in this ball. The conclusion follows from the arbitrariness of the open ball.


    Conversely, choosing any arbitrary point in $\displaystyle B_{\frac{1}{n}}(x)-\{x\}$ finishes the job. There is no need for them to be all distinct.
    Last edited by Drexel28; Jan 27th 2010 at 04:54 PM.
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