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 ??