I'm trying to prove that this inequality holds for all $x, y, z, t \in H$ (H is some Hilbert space):
$\|x-z\|*\|y-t\|\leq \|x-y\|*\|z-t\|+\|y-z\|*\|x-t\|$
My approach is as follows: For all zero vectors it's obvious. Let's assume that none of vectors are zero, then let's make substitution:
$x'=\displaystyle{\frac{x}{\|x\|^{2}}}, t'=\displaystyle{\frac{t}{\|t\|^{2}}}, z'=\displaystyle{\frac{z}{\|z\|^{2}}}$ Let's apply triangle inequality to $\|z'-x'\|:$
$\|z'-x'\|\le\|z'-t'\|+\|t'-x'\| -> \displaystyle{\frac{\|z-x\|}{\|z\|*\|x\|}}\le\displaystyle{\frac{\|z-t\|}{\|z\|*\|t\|}}+\displaystyle{\frac{\|t-x\|}{\|t\|*\|x\|}}$ multiplying both sides of the inequality by $\|z\|*\|x\|*\|y\|*\|t\|$ gives us
$\|z-x\|*\|y\|*\|t\|\le\|z-t\|*\|x\|*\|y\|+\|t-x\|*\|y\|*\|z\|$ and that's the part when I'm stuck.
I have an idea, to show that $\|y-t\|\le\|y\|+\|t\|\le\|y\|*\|t\|$ , but it seems incorrect (at least I don't know any useful inequalities to say that it's true).