$a_n$ is a positive series, and I know that $\limsup a_n \cdot \limsup \frac1{a_n} = 1$.
Prove that $a_n$ is convergent.
What do I need to do?
2026-04-04 01:55:35.1775267735
prove that $a_n$ is convergent if $\limsup a_n \cdot \limsup \frac1{a_n} = 1$
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Can you show that $$\limsup\frac1{a_n}=\frac1{\liminf a_n},$$ (for any sequence $(a_n)$ such that $a_n>0$)?
Once you prove this, the equality you are given simply says that $\limsup a_n=\liminf a_n$.
You can find this also in the book Wieslawa J. Kaczor, Maria T. Nowak: Problems in mathematical analysis: Volume 1; Real Numbers, Sequences and Series as Problems 2.4.22 and 2.4.23. The problems are given on p.45 and solved on p.203-204.