Uniformly continuous integrable dissipation tends to zero ($d\ge 0,{\int }_0^{\infty }d(t)dt<\infty ⟹d(t)\to 0$)

Suppose $d$ is nonnegative, integrable and uniformly continuous on the positive half-line. If $d(t_n)\ge ϵ>0$ at times tending to infinity, uniform continuity supplies intervals of one fixed positive length around those times on which $d\ge ϵ/2$. An infinite disjoint subsequence of these intervals contradicts integrability. Thus the displayed implication holds. Mere smoothness at every finite time is insufficient: smooth peaks of shrinking widths can have finite integral and heights bounded away from zero.