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

  • Thread starter Thread starter Kinetica
  • Start date Start date
  • Tags Tags
    Epsilon
Click For Summary

Homework Help Overview

The discussion revolves around a proof in Real Analysis, specifically addressing the equivalence of the inequality |x-a|< e and the double inequality a-e < x < a+e.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • The original poster attempts to prove the equivalence by assuming each inequality in turn and deriving the other. Some participants offer feedback on the clarity and style of the proof, suggesting ways to improve the presentation.

Discussion Status

Participants are engaged in reviewing the proof attempts, with some expressing agreement on the correctness of the original poster's approach. Suggestions for improving the organization of the proof have been made, indicating a productive exchange of ideas.

Contextual Notes

No specific constraints or missing information have been noted in the discussion.

Kinetica
Messages
83
Reaction score
0

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
 
Physics news on Phys.org
Looks okay to me.
 
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]
 
Thank you, this is a great suggestion. My proofs are indeed full of organizational flaws.
 

Similar threads

Replies
6
Views
3K
  • · Replies 4 ·
Replies
4
Views
2K
  • · Replies 7 ·
Replies
7
Views
2K
  • · Replies 12 ·
Replies
12
Views
2K
  • · Replies 2 ·
Replies
2
Views
2K
  • · Replies 10 ·
Replies
10
Views
3K
Replies
6
Views
2K
Replies
3
Views
1K
  • · Replies 4 ·
Replies
4
Views
2K