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?