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??