Epsilon-Delta Limit in Two Variables

[Vorde]

Homework Statement

Show (using the epsilon-delta method) that the $\lim_{\substack{x\rightarrow a\\y\rightarrow b}} y = b$

Homework Equations

Epsilon-Delta Definition

$\lim_{\substack{x\rightarrow a\\y\rightarrow b}} f(x,y) = L$
means that for every $\epsilon$ > 0, there exists a number $\delta$>0 such that:
$|f(x,y)-L| < \epsilon$ whenever $0 < \sqrt{(x-a)^2+(y-a)^2} < \delta$

The Attempt at a Solution

I totally get what the epsilon-delta definition is and what it means, but I'm not so sure how to go about showing this limit exists.
My textbook has only one example and it's really badly described.

My understanding is that I have
$|y-b| < \epsilon$ when $\sqrt{x^2+y^2} < \delta$
So I can say $y≤\sqrt{x^2+y^2}$
So $|y-b| ≤ \sqrt{x^2+y^2}$?

But I don't get where to go from here. What should my endgoal be?

 

[Robert1986]

So, if \sqrt{(x-a)^2 + (y-b)^2} &lt; \delta, what can you say about how big \sqrt{(y-b)^2} is?

 

[Vorde]

$\sqrt{(y-b)^2} < \sqrt{(y-a)^2+(x-a)^2}$
So
$\sqrt{(y-b)^2} < \delta$
But I don't understand what I'm supposed to show.
Is it enough to say what I just said? I feel like I need more...

 

[Robert1986]

Well, so now you need to pick what your \delta is, right? So, |y-b| = \sqrt{(y-b)^2} &lt; \delta, so what should you pick your \delta to be?

 

[STEMucator]

You know what you want, so apply the definition to show what you want.

\forall ε&gt;0, \exists δ &gt; 0 \space | \space 0 &lt; |x-a|, |y-b| &lt; δ \Rightarrow |y - b| &lt; ε

Now if |y-b| < ε then we are already done. Now choose any arbitrary δ ≤ aε where a \in ℝ, 0 &lt; a &lt; 1 \space as to make the neighborhood around f(x,y) as small or large as we like and still satisfy the definition.

Really depends on how big you want your neighborhood to be right :) ?

 

[Vorde]

Ah! I got it!

Thanks to all of you :)
One less problem left in the world.