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...

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