tag removed - Questions on supremum

If I denote
$$
f(s,t;x)=frac{left|left(1+sright)+txright|+left|left(1+tright)x+sright|}{1+left|xright|+left|s+txright|},
$$
then for $tne0$
$$
fleft(s,t;-frac{s}{t}right)=frac{1+|s/t|}{1+|s/t|}=1,
$$
hence the supremum over all $xinmathbb R$ is at least 1. If $t=0$, then
$$
f(s,0;x)=frac{|1+s|+|x+s|}{1+|s|+|x|}
$$
and
$$
lim_{xtopminfty}f(s,0;x)=1,
$$
so the supremum is (at least) 1 as well.

A similar substitution $x=-t/s$ works for the second supremum.