Proposition 1

Let $(u_n)$ and $(v_n)$ be two series with positive terms. If $u_n \preccurlyeq v_n$ and if the series $(v_n)$ converges, then $(u_n)$ converges; if $u_n \succ v_n$ and if $\sum_{n=1}^{\infty} v_n = +\infty$, then $\sum_{n=1}^{\infty} u_n = +\infty$.