= Solution
Fix $0<c<1$ and choose $R_c$, independently of $X$, with the <normal distribution>
$$
\boxed{R_c\sim\mathcal N\bigl((1-c)\mu,(1-c^2)\sigma^2\bigr).}
$$
The <linear combination of independent normal random variables> $cX+R_c$ is normal. Its <expected value> is $c\mu+(1-c)\mu=\mu$ and its <variance> is $c^2\sigma^2+(1-c^2)\sigma^2=\sigma^2$. Hence $cX+R_c\overset d=X$ for every $c$, proving that every specified <normal distribution> is a <self-decomposable distribution>.
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