Show |x-a|< epsilon IFF a-epsilon < x < a+epsilon

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In summary, the conversation is about a solution to a Real Analysis problem where the goal is to show that |x-a|< e if and only if a-e < x < a+e. The solution provided is deemed acceptable, but it is suggested to improve the style of the proof by using the \iff sign and combining the proofs from each direction.
  • #1
Kinetica
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Homework Statement



Hi! I am new to Real Analysis. Please let me know if my solution is alright. Thanks.
epsilon=e

Show that |x-a|< e IFF a-e < x < a+e

The Attempt at a Solution



Assume |x-a|< e. Prove, a-e < x < a+e

|x-a|< e
-e < x-a < e
a-e < x < a+e

Assume a-e < x < a+e. Prove |x-a|< e.

a-e < x < a+e
-e < x-a < e
|x-a|< e
 
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  • #2
Looks okay to me.
 
  • #3
Looks good to me to, but just to improve the style of your proof: when an operation is reversible, you can use the [itex]\iff[/itex] sign and combine your proofs from each direction into a single proof.

E.g:
[tex] \|x-a\| < \epsilon \iff -\epsilon < x - a < \epsilon \iff a - \epsilon < x < a + \epsilon [/tex]
 
  • #4
Thank you, this is a great suggestion. My proofs are indeed full of organizational flaws.
 

What does the notation "Show |x-a|< epsilon IFF a-epsilon < x < a+epsilon" mean?

This notation is used in mathematics to represent the statement "the absolute value of the difference between x and a is less than epsilon if and only if x is between a minus epsilon and a plus epsilon."

Why is this notation useful?

This notation is useful for proving the convergence of a sequence or the continuity of a function. It allows us to show that as x gets closer to a, the values of x are within a certain range of a.

What is the purpose of the epsilon variable in this notation?

The epsilon variable represents a small positive number that can be chosen arbitrarily small. It is used to show that the values of x are getting closer and closer to a.

Can you give an example of how this notation is used in a proof?

Sure, let's say we want to prove that the sequence (1/n) converges to 0 as n approaches infinity. We can use this notation to show that for any given epsilon, there exists a value of n such that 0 < 1/n < epsilon. This proves that the sequence is getting closer and closer to 0 as n increases.

Is this notation specific to a certain branch of science or can it be applied to various fields?

This notation is not specific to a certain branch of science and can be applied to various fields, such as mathematics, physics, engineering, and computer science. It is a fundamental concept in calculus and is used in many different areas of science and research.

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