You agree that , right?
then let and .
Note that .
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
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
limsup(x_n+y_n)+limsup (-x_n)>=limsup y_n
it should be equal sign
why its bigger ??