- #1
spenghali
- 14
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Homework Statement
If limsup(an) and limsup(bn) are finite, prove that limsup(an+bn) [tex]\leq[/tex] limsup(an) + limsup(bn).
Homework Equations
The Attempt at a Solution
My proof seems a bit short, so if someone could please reassure me this is a valid proof, thanks in advance.
Proof: Assuming an and bn are bounded sequence. Let a > limsup(an) and b > limsup(bn). Then a+b > an+bn for all but finitely many n's. This implies that a+b [tex]\geq[/tex] limsup(an+bn). Since this hold for any a [tex]\geq[/tex] limsup(an) and any b > limsup(bn), this implies limsup(an+bn) [tex]\leq[/tex] limsup(an) + limsup(bn). QED