 Real-axis singular limit of the Riemann zeta function at one

For $p>0$, integral comparison for $x^{-1-p}$ bounds its convergent positive series by $1/p\le {\sum }_{n\ge 1}{n}^{-1-p}\le 1+1/p$. Multiplication by $p$ gives $1\le p\zeta (1+p)\le 1+p$, so the squeeze theorem proves the displayed real-axis limit. This establishes the leading $1/p$ divergence by real integration alone. The convergence of the series at each fixed positive $p$ follows from bounded increasing partial sums via the monotone convergence theorem for sequences.