Analysis Problem for homework, infimum and supremum

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

The easy inequality follows from

[tex]\sup_n{a_n+b_n}\leq \sup_n{a_n}+\sup_n{b_n} [/tex]

just take the limit of both sides.

For the other inequality, suppose that

[tex]\limsup{a_n+b_n}<s+\limsup{b_n}[/tex]

Then there exist an [tex]\epsilon>0[/tex] such that

[tex]\limsup{a_n+b_n}<\limsup{b_n}+s-\epsilon:=L[/tex]

Since [tex]\limsup{a_n+b_n}<L[/tex], it follows that

[tex]\exists n_0:~\forall n>n_0:~a_n+b_n<L [/tex]

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

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

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

[tex]\l
 

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