This article is devoted to sequences $(u_{n})_{n}$ of subharmonic functions in $\mathbb{R}^{N}$, with finite order, whose means $J_{u_{n}}(r)$ (over spheres centered at the origin, with radius r) satisfy such a condition as: $\forall r > 0$, $\exists A_{r} > 0$ such that $J_{u_{n}}(r) \le A_{r}$, $\forall n \in \mathbf{N}$. The paper investigates under which conditions one may extract a pointwise or uniformly convergent subsequence.