Clipping preserves divergence of a positive series (${\sum }_n{x}_n=\infty ⟹{\sum }_n\min (x_n,c)=\infty$)

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.