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Math Help - Simple proving - need some help

  1. #1
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    Simple proving - need some help

    a) Let I = {x E R : x^3 < 2}. Show that I is a left infinite interval.

    Now I understand that say x is between -infinity and a.

    It also makes sense as when x is negative, x^3 is always negative.

    But I don't know how to PROVE it using small steps.

    Any advice/help?
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  2. #2
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by chr91 View Post
    a) Let I = {x E R : x^3 < 2}. Show that I is a left infinite interval.

    Now I understand that say x is between -infinity and a.

    It also makes sense as when x is negative, x^3 is always negative.

    But I don't know how to PROVE it using small steps.

    Any advice/help?
    \displaystyle x^3 < 2 \Longleftrightarrow x < \sqrt[3] 2 (since x^3 is an increasing function), that is, \displaystyle x \in (-\infty , \sqrt[3] 2)
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  3. #3
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    Quote Originally Posted by Jhevon View Post
    \displaystyle x^3 < 2 \Longleftrightarrow x < \sqrt[3] 2 (since x^3 is an increasing function), that is, \displaystyle x \in (-\infty , \sqrt[3] 2)
    I don't think that's the type of answer it's looking for. We are still in the basic step by step proving by axiom stage.

    Surely you've just stated that x is a left infinite interval and then found the right interval?

    How do we actually show that it is a left infinite interval without just saying it or listing negative numbers to prove it?
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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by chr91 View Post
    I don't think that's the type of answer it's looking for. We are still in the basic step by step proving by axiom stage.

    Surely you've just stated that x is a left infinite interval and then found the right interval?

    How do we actually show that it is a left infinite interval without just saying it or listing negative numbers to prove it?
    ...my answer is a left infinite interval. on the number line, it is to the left of  \sqrt[3] 2. and i did show it without just saying it or listing numbers.

    And my first step is pretty basic. it should be covered by your axioms. if you are concerned about that, then it is best for you to state what axioms you can use so that we can answer in the right context, no?
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    Quote Originally Posted by chr91 View Post
    I don't think that's the type of answer it's looking for. We are still in the basic step by step proving by axiom stage.

    Surely you've just stated that x is a left infinite interval and then found the right interval?

    How do we actually show that it is a left infinite interval without just saying it or listing negative numbers to prove it?

    Show that I have no minimum or maximum, show that inf{I} isn't exist and show that sup{I}=\sqrt[3](2) ( use epsilon for all showings...).
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