Cant understand what are they doing in this part of a solution

[transgalactic]

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 ??

 

[phreak]

Because in general, they're not equal. For instance, let x_n = 0 if n even, 1 if n odd. Then lim sup (-x_n) = 0, and lim sup(x_n) = 1. Now let y_n = 0 for all n. Then we have 1 = lim sup (x_n+y_n) + lim sup(-x_n) > lim sup y_n = 0.