Math Help - cant understand what are they doing in this part of a solution..

1. cant understand what are they doing in this part of a solution..

x_n and y_n are bounded
the first part was proved
we have that
limsup x_n+limsup y_n>=limsup(x_n+y_n)
now we need to prove that:
limsup(x_n+y_n)>=liminf x_n +limsup y_n

as n->infinity
the say:
limsup(-x_n)=-liminf(x_n)

then they say
limsup(x_n+y_n)-liminf x_n=limsup(x_n+y_n)+limsup (-x_n)>=limsup y_n

so by putting liminf x_n on the other side we get
imsup(x_n+y_n)>=liminf x_n +limsup y_n

why
limsup(x_n+y_n)+limsup (-x_n)>=limsup y_n

it should be equal sign
why its bigger ??

2. You agree that $\limsup a_n+\limsup b_n\ge \limsup (a_n+b_n)$, right?

then let $a_n=x_n+y_n$ and $b_n=-x_n$.

Note that $a_n+b_n=y_n$.