= Uniformly continuous integrable dissipation tends to zero
{title2=$d\geq0,\quad \int_0^\infty d(t)\,dt<\infty\ \Longrightarrow\ d(t)\to0$}
Suppose $d$ is nonnegative, integrable and <uniformly continuous> on the positive half-line. If $d(t_n)\geq\epsilon>0$ at times tending to infinity, <uniform continuity> supplies intervals of one fixed positive length around those times on which $d\geq\epsilon/2$. 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.
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