Prove that there is no positive real

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Homework Help Overview

The problem involves proving that there is no smallest positive real number, exploring concepts related to real numbers and their properties.

Discussion Character

  • Exploratory, Conceptual clarification

Approaches and Questions Raised

  • The original poster attempts a proof by contradiction, suggesting that if a smallest positive real number exists, then a smaller positive number can be found. Other participants question the need for additional rigor in proving the positivity and ordering of the midpoint.

Discussion Status

The discussion is active, with participants providing feedback on the original proof attempt and suggesting areas for further clarification. There is no explicit consensus on the necessity of additional steps, but guidance has been offered regarding the proof's rigor.

Contextual Notes

Participants are considering the level of rigor required for the proof, with some suggesting that proving the positivity of the midpoint may be trivial.

cragar
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Homework Statement


Prove that there exists no smallest positive real number.

The Attempt at a Solution


Lets assume for contradiction that x is the smallest positive real.
Now we will look at the midpoint between 0 and x which is x/2, well x/2 is positive and smaller than x so this is a contradiction, so there is no smallest positive real.
 
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Looks fine to me. If you want to be pedantic, you may want to prove explicitly that ##x/2## is positive and strictly less than ##x##.
 
Depending on how much rigor you need to put in, you might want to prove ##0<\frac{x}{2}<x##. I know it's trivial, so perhaps it's not needed.
 
ok thanks, it almost seemed to easy just wanted to make sure it worked.
 

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