Hi all, I'm trying to understand one of the proofs in my multivariable analysis class. In this proof, we are relating iterated limits with function limits (not sure if that's the right terminology). In this proof, we use $\delta / \sqrt 2$ to show that the iterated limits exist. However, I'm not exactly sure why this is done. I understand that it probably will allow us to conclude that $$\text{something} < \delta \Rightarrow \text{something else} < \epsilon$$ but I'd like to know how exactly that happens. I assume that this step is probably hidden in the "It follows that" part. Could someone expand on that?
$$\|(x,y)-(a,b)\|=\sqrt{(x-a)^2+(y-b)^2}\lt\sqrt{\left({\delta \over \sqrt{2}}\right)^2+\left({\delta \over \sqrt{2}}\right)^2}=\delta$$. From the definition of
we know that for a given $\varepsilon$ there is a $\delta$ such that if $\|(x,y)-(a,b)\|<\delta$ we have 
$\varepsilon$
So selecting 
and 
ensures 
$\varepsilon$
In a comment Jyrki Lahtonen in a comment to the question already stated that $\delta \over \sqrt{2}$ is not the only possibility, for example you can use
$\delta \over 2$ and get
$$\|(x,y)-(a,b)\|=\sqrt{(x-a)^2+(y-b)^2}\lt\sqrt{\left({\delta \over {2}}\right)^2+\left({\delta \over {2}}\right)^2}= {\delta \over 2} \lt \delta$$