Clipping preserves divergence of a positive series
= Clipping preserves divergence of a positive series
{title2=$\sum_nx_n=\infty\implies\sum_n\min(x_n,c)=\infty$}
If $x_n>0$ and $c>0$, clipping every term at $c$ preserves divergence. If infinitely many terms are at least $c$, the clipped series has infinitely many terms equal to $c$ and diverges. Otherwise its tail agrees with the original series. This lemma is useful in proving convergence from minimum-decrement inequalities.