$\zeta$ has a simple pole of residue $1$ at $s=1$: boundedness of $\zeta(s) - (s-1)^{-1}$ · Prove2Me

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$\zeta$ has a simple pole of residue $1$ at $s=1$: boundedness of $\zeta(s) - (s-1)^{-1}$ · Prove2Me