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From Dirichlet’s theorem we get $f$ is bounded, and hence $\lim_{n\rightarrow \infty}f(\frac 1n)=0$. How?
Let $f$ be a function on the positive integers such that $f(n)>0$ for all $n>0$.
(1)Show that $$\lim_{n\rightarrow \infty}f(\frac 1n)=0.$$
(2)Show that if $f(n)>0$ for all $n>0$ and if $\sum_{n=1}^{\infty}f(n)$ is finite, then $f(n)=0$ for all sufficiently large $n$.
I know that the first part is false if we assume $f(n)>0$ for all $n$, but I don’t know how we get $\lim_{n\rightarrow \infty}f(\frac 1n)=0$ from here.
A:
If $L=\displaystyle\lim_{x\to\infty} f(x)$ then by the Squeeze Theorem, for any $\epsilon>0$, there exists $M$ such that
$$L-\epsilonM$, we have
$$0
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Anonymous is an American thriller drama television series created by Salim Akhtar that was ordered straight-to-series at The CW on September 22, 2018. Based on the Japanese manga series, the series is co-produced by Warner Bros. Television and Platinum Dunes and centers on a team of investigators with psychologically-based abilities who operate out of anonymity.
On November 15, 2018, The CW officially ordered the series to series. On May 10, 2019, The CW canceled the series after one season, citing a viewership of approximately 900,000 viewers.
Premise
The series explores how a team of investigators with psychologically-based abilities work in secret from law enforcement agencies in order to stop violent crimes and catch criminals.
Cast
Main
Aimee Garcia as Paige Matthews, a 27-year-old FBI agent
Richard Harmon as Sam Braun, an FBI agent
Ryan McPartlin as Mike Weston, an FBI agent
Stephanie Chirico as Bree, an FBI agent
Matt Lauria as David Kim, FBI agent
Recurring
Sara Chalke as Rebecca Brown
Chris Tardio as Chad Lake
Franklin J. Alvarez as Felix Maldonado
Jon Huertas as Joe
Guest stars
Ben Bassatt as Paul Flanner
Cindy Chastain as Kiana Edmondson
Jason Clarke as Walker Cobb
Neal McDonough as FBI agent
Rhys Morgan as Allison
Camille Guaty as Agent Sanford.
James Jarecke as Agent McConnell
Maia Mitchell as Charlie Cummings
Jessica Szhor as Allison Nixon
Episodes
Production
Development
On May 10, 2019, The CW canceled the series after one season, citing a viewership of approximately 900,000 viewers.
Casting
On May 10, 2019, The CW canceled the series after one season, citing a viewership of approximately 900,000 viewers
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