= Solution
The PDF bounds the \i[absolute value] of the forcing. Taking absolute values in the <Green function> <integral> and using $|\sin|\leq1$ yields
$$
|x(t)|\leq\frac C\mu\int_0^te^{-(t-s)}e^{-as}\,ds.
$$
For $a\neq1$, this gives
$$
\boxed{|x(t)|\leq\frac C\mu\frac{e^{-at}-e^{-t}}{1-a}\longrightarrow0.}
$$
The ratio is nonnegative for either $a<1$ or $a>1$. The missing special case is $a=1$, for which
$$
\boxed{|x(t)|\leq\frac C\mu te^{-t}\longrightarrow0.}
$$
This proves <decay of a damped oscillator under exponentially decaying forcing>. A one-sided upper bound on $g$ without an absolute value, as in the transcription, would not justify the conclusion.
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