Thread: Epsilon - Delta Proof with Squares

Hey i've been working on this problem for a few days now, just seeing if anyone else has some insight.

∀x ∈ R, ∀ε ∈ R+, ∃δ ∈ R+, ∀y ∈ R, |x – y| < δ ==> |x^2 – y^2| < ε

prove using epsilon-delta structured proof

ive got to the point where

|x^2 – y^2| < δ|x + y|

but am not completely stuck, I think I have to replace y with a constant somehow? any help would be great! thanks in advance!

I cannot read that very well. It also would help if you would add the entire question.

You mentioned that
and

From there I would rewrite




Then


To find C, I would look at:
if then


In this case y would be a

But to be sure I would need the entire question. If you still need this problem, maybe you can submit the question asked.

Originally Posted by kale3193

Hey i've been working on this problem for a few days now, just seeing if anyone else has some insight.

∀x ∈ R, ∀ε ∈ R+, ∃δ ∈ R+, ∀y ∈ R, |x – y| < δ ==> |x^2 – y^2| < ε

prove using epsilon-delta structured proof

ive got to the point where

|x^2 – y^2| < δ|x + y|

but am not completely stuck, I think I have to replace y with a constant somehow? any help would be great! thanks in advance!

if i am not wrong this is just the prove that f(x) = x^2 is continuous for all x on R.
if we take delta = min{ |x|/2, 2epsilon/5|x| } for all x from R \ {0} the formula will work

and for x = 0 we can take delta = sqrt( epsilon )

you choose x and epsilon first and than you choose delta, so delta must depend just on x and epsilon, not on y

How can you write < epsilon first?
I thought we had to assume |x-y| < delta first and show that it leads to |x^2 - y^2 | < Epsilon

Originally Posted by kale3193

How can you write < epsilon first?
I thought we had to assume |x-y| < delta first and show that it leads to |x^2 - y^2 | < Epsilon

i didnt wrote < epsilon first...
you need to find delta ( based on epsilon and x ) such that the statment in first post is true! Dont know if you learnd about continuosity of functions ( maybe you did, but didnt realise that very good, because this is just an example of that )