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Math Help - interval on real number line

  1. #1
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    interval on real number line

    Suppose that I have an half open interval, [1, 5+\frac{1}{n}).

    Is it correct to say that the half open interval will become a close interval, [0,5] as n approaches  \infty ?

    My thinking is that \frac{1}{n} to the right of the number 5 changes, but it could never affect the number 5. In essence, at the end the interval will be [1, 5+\epsilon). If I drop the \epsilon, I will still get [0,5].

    I am not too sure. I need help.
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  2. #2
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    Yes, it's true. For any number larger than 5 (call it 'x'), you can always find a value of n such that x is not in the interval [1,5+1/n). Therefore, it becomes a closed interval [1,5].

    Another way to say this is:

    \bigcap_{n=1}^{\infty} ~ \left[1,5+\frac{1}{n}\right) = [1,5]
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  3. #3
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    Quote Originally Posted by drumist View Post
    Yes, it's true. For any number larger than 5 (call it 'x'), you can always find a value of n such that x is not in the interval [1,5+1/n). Therefore, it becomes a closed interval [1,5].

    Another way to say this is:

    \bigcap_{n=1}^{\infty} ~ \left[1,5+\frac{1}{n}\right) = [1,5]
    Thank you for giving the logical explanation.
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