= Solution
The claimed endpoint bounds with radius $\epsilon$ are \b[false for positive corner perturbations]. For example, take $n=3$, $\epsilon=1/1000$ and $\delta=9/10<1-\epsilon$. The characteristic <polynomial> is
$$
P(\lambda)=(\lambda-1)(\lambda-2)(\lambda-3)
-\frac1{4000}(2\lambda-4)-\frac9{160}.
$$
Throughout $|\lambda-1|\le1/1000$, the absolute value of the sum of its first two terms is below $0.003$, whereas $9/160=0.05625$. Hence this interval contains no root, contradicting the first requested bound. The actual <eigenvalues> are approximately $1.029144,1.943599,3.027257$, which also violate the last requested bound.
There is a valid real-eigenvalue conclusion with wider endpoint bounds. For $0<\epsilon<1/4$ and $\delta<(n-1)/2-\epsilon$, the <Gershgorin discs> from rows have one enlarged component on the left; the <Gershgorin discs> from columns have one enlarged component on the right. Each has an enlarged radius $r=\delta+\epsilon<(n-1)/2$. A small disc that meets the enlarged left disc has index $j$ satisfying $j-1\le r+q<n/2$. A small disc meeting the enlarged right disc satisfies $n-j<n/2$. Thus, when $n=2m$, every disc with index at most $m$ is isolated in the column family and every disc with index at least $m+1$ is isolated in the row family. Their counts give all $n$ <eigenvalues>, one in each disjoint small disc, so all are real.
When $n=2m+1$, the same argument isolates the first $m$ discs in the column family and the last $m$ in the row family. It accounts for $2m$ real <eigenvalues>. The one remaining <eigenvalue> is real by conjugation, and belongs to the central small disc: the enlarged left and right discs are disjoint, and their two components can intersect only within the disc centered at $m+1$. Therefore the valid uniform bounds are
$$
\boxed{|\lambda_1-1|\le\tfrac14,\quad
|\lambda_i-i|\le\tfrac14+\epsilon\ (2\le i<n),\quad
|\lambda_n-n|\le\tfrac14.}
$$
The sharper endpoint radii $\epsilon$ hold in part (d) when $\delta=0$, and cannot simply be carried over to the perturbed <matrix>.
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