Math Help - lim inf/sup innequality question...

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??

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??

in the solution they say that
because (x_r_n +y_r_n) is convergent then it equals limsup(x_n +y_n)

so what that it is a convergent?