Monotonicity of uniform sparse recovery in the exponent

If the Lq null space property holds at $q\le 1$, it holds at every $0<p<q$. Order a nonzero null vector's magnitudes $a_1\ge \cdots \ge a_N$ and put $t=a_s>0$. Since $p-q<0$, the top-$s$ $p$-power sum is at most $t^{p-q}$ times its $q$-power sum, while the tail $p$-power sum is at least that multiple of the tail $q$-power sum. The strict $q$ inequality therefore gives the strict $p$ inequality. The largest $s$ magnitudes are the worst support of a vector, so all other supports satisfy it too.