# Lim inf/sup innequality question

1. Mar 12, 2009

### transgalactic

x_n and y_n are bounded
$$lim sup x_n+lim sup y_n>=lim x_r_n+lim y_r_n$$
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
$$lim(x_r_n+y_r_n)=limsup(x_n+y_n)$$
this is true because the sequence is constructed from a convergent to the sup sub sequences
so they equal the lim sup of x_n+y_n

now i need to prove that
$$limsup(x_n+y_n)=>liminf x_n+limsup y_n$$

i tried:
$$limsup(x_n+y_n)=limsup x_n+limsup y_n=>liminf x_n+limsup y_n$$
lim sup i always bigger then lim inf

so its true

did i solved it correctly?

Last edited: Mar 12, 2009