Uniform error bound for nearby decaying exponentials (${\sup }_{x\ge 0}|e^{-ax}-e^{-bx}|\le |a-b|/[e\min (a,b)]$)

For positive rates $a,b$, the mean value theorem in the rate gives a bound by $|a-b|x{e}^{-\min (a,b)x}$. Its maximum is $|a-b|/[e\min (a,b)]$. Combining this estimate with coefficient errors bounds a finite spectral sum uniformly on a half-line, including slow scales much longer than one. Positivity of both rates is essential: unstable exponentials need a different time-window or relative-error analysis.