x_r_n is a subsequence of x_n
y_r_n is a subsequence of y_n
lim(x_r_n+y_r_n)=limsup(x_n +y_n)
how they got to this conclusion??
why this is true
x_n and y_n are bounded (as n->infinity)
lim sup x_n >=lim x_r_n
lim sup y_n >=lim y_r_n
lim sup x_n+lim sup y_n>=lim x_r_n+lim y_r_n=lim(x_r_n+y_r_n)
on what basis they get to lim(x_r_n+y_r_n) from sum of limits??
and then
lim(x_r_n+y_r_n)=limsup(x_n +y_n)
how they got to this conclusion??