We consider subordinators $X_α=(X_α(t))_{t\ge 0}$ in the domain of attraction at 0 of a stable subordinator $(S_α(t))_{t\ge 0}$ (where $α\in(0,1)$); thus, with the property that $\overlineΠ_α$, the tail function of the canonical measure of $X_α$, is regularly varying of index $-α\in (-1,0)$ as $x\downarrow 0$. We also analyse the boundary case, $α=0$, when $\overlineΠ_α$ is slowly varying at 0. When $α\in(0,1)$, we show that $(t \overlineΠ_α(X_α(t)))^{-1}$ converges in distribution, as $t\downarrow 0$, to the random variable $(S_α(1))^α$. This latter random variable, as a function of $α$, converges in distribution as $α\downarrow 0$ to the inverse of an exponential random variable. We prove these convergences, also generalised to functional versions (convergence in $\mathbb{D}[0,1]$), and to trimmed versions, whereby a fixed number of its largest jumps up to a specified time are subtracted from a process. The $α=0$ case produces convergence to an extremal process constructed from ordered jumps of a Cauchy subordinator. Our results generalise random walk and stable process results of Darling, Cressie, Kasahara, Kotani and Watanabe.