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I am reading the paper Dirichlet's theorem: a real variable approach by Robin Chapman. In this paper, he constructs a proof via real analysis rather than complex analysis that $\zeta(s)$ is convergent if and only if $s>1$. However, this is a standard fact known about $\zeta(s)$. What confuses me is this:

He states as a consequence of the inequality $\frac{s}{s-1} > \zeta(s) > \frac{1}{s-1},$ the following limit is true: $\lim_{s \to 1^{+}}(s-1)\zeta(s)=1.$ I know this is probably a stupid question, but I'm not that great with limits. I can't quite see where this reasoning is derived from. Is this the case because of the equivalent inequality $s>(s-1)\zeta(s)>1$ where $s >1$? If so, how?

Could anyone care to elucidate this rudimentary step in logic for me?

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    @EmileOkada, please restore you a$n$swer so that I ca$n$ accept it. :)2012-04-19

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This follows from the squeeze theorem. $\frac{s}{s-1}>\zeta(s)>\frac{1}{s-1}$ $\frac{s(s-1)}{s-1}>\zeta(s)(s-1)>\frac{1(s-1)}{s-1}$ $s>\zeta(s)(s-1)>1$ Since $\lim_{s\rightarrow1^+}s=\lim_{s\rightarrow1^+}1=1$ $\lim_{s\rightarrow1^+}\zeta(s)(s-1)=1$