= Solution
<Subadditivity of Von Neumann entropy> applied to $RE$ gives $S(RE)\leq S(R)+S(E)$. Together with the environment expression for <coherent information>,
$$
I_c(\Lambda_1,\rho)=S(RE)-S(E)\leq S(R).
$$
The channel does not act on $R$. Its reduced state has the same nonzero eigenvalues as the original input $\rho$, because $RA$ initially purifies that input. Hence
$$
\boxed{I_c(\Lambda_1,\rho)\leq S(\rho).}
$$
More precisely, $I_c=S(\rho)-I(R:E)$: the <coherent information upper bound by input entropy> is a consequence of nonnegative <quantum mutual information> with the environment. Equality holds when $R$ and $E$ are uncorrelated.
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