Analysis Problem for homework, infimum and supremum

[retspool]

I have this analysis homework due tomorrow.
This is one of my problems.Let (sn) and (tn) be sequences in R. Assume that lim sn = s ∈ R. Then lim sup(sn +tn) = s+limsup(tn).I don't even know how to approach it. Even though it seems very straight forward.

 

[micromass]

You will have to prove 2 inequalities, a hard one and an easy one.

The easy inequality follows from

\sup_n{a_n+b_n}\leq \sup_n{a_n}+\sup_n{b_n}

just take the limit of both sides.

For the other inequality, suppose that

\limsup{a_n+b_n}<s+\limsup{b_n}

Then there exist an \epsilon>0 such that

\limsup{a_n+b_n}<\limsup{b_n}+s-\epsilon:=L

Since \limsup{a_n+b_n}<L, it follows that

\exists n_0:~\forall n>n_0:~a_n+b_n<L

But from a certain n_1 it is true that a_n-s>-\epsilon/2. Thus from \max\{n_0,n_1\}, we must have that

s-\epsilon/2+b_n<a_n+b_n<L=\limsup{b_n}+s-\epsilon

So from a certain moment, we have that b_n<\limsup{b_n}-\epsilon/2. But the definition of limsup tells us that this is impossible...

\l