Epsilon-Delta Proof Homework: Understanding the Thought Process

[dainty77]

Homework Statement

For every ε> 0, there is a δ> 0 such that 1- δ< x <1 + implies 2- ε <7-5x <2 + ε

Homework Equations

The Attempt at a Solution

My understanding of epsilon-delta proofs is very minimal at this point. Was hoping someone would be able to explain your thought process when attempting these kinds of proofs. Thank you!

 

[tiny-tim]

hi dainty77!

Was hoping someone would be able to explain your thought process when attempting these kinds of proofs.

my thought process would be to attempt the simplest possible solution first …

ie try δ = kε first (for a constant k),

then δ = kε², then δ = k√ε and so on

 

[HallsofIvy]

I would start with 2- \epsilon&lt; 7- 5x and see that -5- \epsilon&lt; -5x so that x&lt; 1+ \epsilon/5 and x- 1&lt; \epsilon/5. Then turn to 7- 5x&lt; 2+ \epsilon, so that -5x&lt; -5+ \epsilon and x&gt; 1- \epsilon/5, x- 1&gt; -\epsilon/5. Now do you see what \delta must be?

 

[dainty77]

hi dainty77!


my thought process would be to attempt the simplest possible solution first …

ie try δ = kε first (for a constant k),

then δ = kε², then δ = k√ε and so on

I see where you are gettiing at! Let me work on it some more. Thank you for your help!