= Solution
For fixed $p>0$, the partial sums are increasing and bounded above by $1+1/p$. The <monotone convergence theorem for sequences> therefore proves convergence to a finite sum $S(p)$. Passing to the limit $N\to\infty$ in the two bounds gives
$$
\frac1p\leq S(p)\leq\frac1p+1,\qquad1\leq pS(p)\leq1+p.
$$
The <squeeze theorem> now yields
$$
\boxed{\lim_{p\downarrow0}p\sum_{n=1}^\infty n^{-1-p}=1.}
$$
Since $S(p)=\zeta(1+p)$, this is the <real-axis singular limit of the Riemann zeta function at one>. The limiting sum is taken at fixed $p$ before sending $p$ to zero; setting $p=0$ inside the convergent-series argument would instead give the divergent harmonic series.
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