Vtgyjfy

Let $x_n$ and $y_n$ denote sequences such that:
$$
lim_{ntoinfty}(|x_n + y_n| - |x_n - y_n|) = +infty
$$
Prove:
$$
lim_{ntoinfty}(|x_n + y_n| - |x_n - y_n|) = +infty iff lim_{ntoinfty}|x_n| =lim_{ntoinfty}|y_n| =lim_{ntoinfty}x_ny_n =+infty
$$

I've started with the first case ($implies$):
$$
lim_{ntoinfty}(|x_n + y_n| - |x_n - y_n|) = +infty \
stackrel{text{def}}{iff} forall epsilon > 0 exists NinBbb N: forall n>N implies ||x_n + y_n| - |x_n - y_n|| ge epsilon
$$

We want to show that $|x_n| ge epsilon$ and $|y_n|ge epsilon$. By triangular inequality:
$$
|x_n + y_n| + |x_n - y_n| ge ||x_n + y_n| - |x_n - y_n|| ge epsilon
$$

At the same time:
$$
|x_n + y_n| + |x_n - y_n| ge |(x_n + y_n) + (x_n - y_n)| = 2|x_n|
$$
Or:
$$
|x_n + y_n| + |y_n - x_n| ge |(x_n + y_n) + (y_n - x_n)| = 2|y_n|
$$
Here is where the first questionable case comes in. It seems that:
$$
|x_n + y_n| + |x_n - y_n| ge 2|x_n| ge ||x_n + y_n| - |x_n - y_n|| ge epsilon\
text{and}\
|x_n + y_n| + |y_n - x_n| ge 2|y_n| ge ||x_n + y_n| - |x_n - y_n|| ge epsilon tag1
$$

But I'm not sure how to justify that. I've checked various cases like $x<0 land y<0$ and 3 others, for all it seems to hold. So based on this:
$$
forall epsilon >0 exists NinBbb N: forall n> N implies 2|x_n| ge epsilon \
forall epsilon >0 exists NinBbb N: forall n> N implies 2|y_n| ge epsilon
$$

Which shows:
$$
lim_{ntoinfty}|x_n| = lim_{ntoinfty}|y_n| = +infty
$$

In this part I'm interested in how to justify $(1)$ and where from it follows that:
$$
lim_{ntoinfty}x_ny_n = +infty
$$

Second case $(impliedby)$. This case I've no idea where to start from. We basically given three things:
$$
forall epsilon > 0 exists NinBbb N: forall n>N implies |x_n| ge epsilon\
forall epsilon > 0 exists NinBbb N: forall n>N implies |y_n| ge epsilon \
forall epsilon > 0 exists NinBbb N: forall n>N implies |x_ny_n| ge epsilon
$$

The problem is I don't see where to go from this.

Could you please verify the overall reasoning and help with ($impliedby$) and question from ($implies$), I still often have troubles with constructing those proofs since i'm a self-learner and have no one to refer to. Thank you!

$(Longrightarrow)$ Note that by triangle inequality, we have
$$
2|x_n|=|(x_n+y_n)+(x_n-y_n)|ge|x_n+y_n|-|x_n-y_n|
$$ and
$$
2|y_n|=|(x_n+y_n)-(x_n-y_n)|ge|x_n+y_n|-|x_n-y_n|.
$$ By taking $ntoinfty$, we get
$$
lim_{ntoinfty}|x_n|=lim_{ntoinfty}|y_n|=infty.
$$ Also note that
$
|x_n+y_n|ge |x_n-y_n|
$ holds eventually. Hence, by squaring both sides, we have
$x_ny_nge 0$. It follows that $lim_{ntoinfty}x_n y_n=lim_{ntoinfty}|x_n| |y_n|=infty.$

$(Longleftarrow)$ To show the converse, note that $lim_{ntoinfty}x_n y_n=infty$ implies that $$
|x_n+y_n|-|x_n-y_n|ge 0
$$ eventually. And since it is a positive sequence, we have
$$
lim_{ntoinfty}|x_n+y_n|-|x_n-y_n|=infty Longleftrightarrow lim_{ntoinfty}left[|x_n+y_n|-|x_n-y_n|right]^2=infty.
$$ Now, we have
$$
left[|x_n+y_n|-|x_n-y_n|right]^2= 2(|x_n|^2+|y_n|^2) -2|x_n^2-y_n^2|=:L.
$$ Observe that if $x_n^2ge y_n^2$, then it holds $L=4|y_n|^2$ and otherwise, $L=4|x_n|^2$. This gives $L=4 min{|x_n|^2,|y_n|^2}$ and
$$
lim_{ntoinfty}left[|x_n+y_n|-|x_n-y_n|right]^2= 4lim_{ntoinfty}min{|x_n|^2,|y_n|^2}=infty
$$ by the assumption that $lim_{ntoinfty}|x_n|=lim_{ntoinfty}|y_n|=infty$. This proves the converse claim.