x_n and y_n are bounded
this is true because there presented sub sequences converge to the upper bound of x_n and y_n .
the sum of limits is the limit of sums so we get one limit
and the of two convergent sequence is one convergent sequence
this is true because the sequence is constructed from a convergent to the sup sub sequences
so they equal the lim sup of
now i need to prove that
lim sup i always bigger then lim inf
so its true
did i solved it correctly?