= Solution
Since $0<1-2\epsilon<1$, the off-diagonal multiplier of the <iterated phase-flip channel> tends to zero. Thus
$$
\boxed{\lim_{n\to\infty}D_{\epsilon_n}(\rho)=\frac{\rho+Z\rho Z}{2}
=\begin{pmatrix}u&0\\0&1-u\end{pmatrix}.}
$$
Let $P_0=|0\rangle\langle0|$ and $P_1=|1\rangle\langle1|$ be the <computational basis> measurement projectors. The <nonselective projective measurement> channel is
$$
P_0\rho P_0+P_1\rho P_1
=u|0\rangle\langle0|+(1-u)|1\rangle\langle1|,
$$
which is exactly the displayed limit. Its diagonal entries retain the original outcome <probabilities>; the measurement outcome is ignored and the coherences are erased. The limiting evolution is therefore the <completely dephasing channel>.
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