The Harmonic Series Diverges

In the setting of The Real Numbers: Standing Notation and Background, natural numbers $m$ are read in $\mathbb{R}$ through the canonical map fixed there, and convergence of a series of real numbers is as defined there.¶ The set $\bigl\{\sum_{m=1}^{M}\tfrac{1}{m}:M\in\mathbb{N}\bigr\}$ is not bounded above; in particular the series $\sum_{m=1}^{\infty}\tfrac1m$ does not converge.