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

i tried:

lim sup i always bigger then lim inf

so its true

did i solved it correctly?