= Solution
\b[The endpoint bounds printed in this part are false.] Take $n=3$, $\epsilon=1/100$ and $\delta=9/10$, which obey $\delta<(n-1)/2-\epsilon=99/100$. The <characteristic polynomial> is
$$
P(z)=z^3-6z^2+\frac{2199}{200}z-\frac{4837}{800}.
$$
It has $P(101/100)=-31599/10^6<0$ and $P(5/4)=441/1600>0$, so it has a real root strictly between $1.01$ and $1.25$, outside the first printed interval. There is a unique <eigenvalue> in the transpose's isolated first disc of radius $1/4$, identifying it as the first <eigenvalue>. Similarly $P(301/100)=-40999/10^6<0$ and $P(13/4)=41/64>0$ give a root above $3.01$, violating the last printed interval. The approximate three roots are $1.02675615$, $1.94385374$ and $3.02939011$.
There is nevertheless a corrected enclosure proving the claimed reality. Under the stated condition write $R=\epsilon+\delta<(n-1)/2$. The large first row disc is disjoint from the large last column disc, since their centres are $n-1$ apart and their radii are both $R$. Every <eigenvalue> lies both in the row union and in the column union. It cannot belong to both large discs, so it lies in at least one of the fixed small discs
$$
S_1:\ |z-1|\le1/4,\qquad S_i:\ |z-i|\le1/4+\epsilon\ (2\le i\le n-1),\qquad S_n:\ |z-n|\le1/4.
$$
These fixed discs are pairwise disjoint. At $\delta=0$ each contains one <eigenvalue>. Eigenvalues depend continuously, as a multiset, on the entries; throughout the <connected> allowed interval no <eigenvalue> can move between these separated fixed sets while remaining in their union. Thus each still contains one, counted with multiplicity. Complex-conjugate pairing now proves each is real. The valid conclusion is
$$
\boxed{|\lambda_1-1|\le1/4,\quad|\lambda_i-i|\le1/4+\epsilon\ (2\le i\le n-1),\quad|\lambda_n-n|\le1/4}.
$$
The source's stronger width $\epsilon$ at both endpoints does not persist after adding the corner entry; it was valid for the unperturbed matrix only.
The enclosure method is <Complementary row and column Gershgorin isolation>.
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