Solution
= Solution
The transpose has first radius $1/4$, interior radii $1/4+\epsilon$, and last radius $\epsilon$. These discs are again disjoint and each has one <eigenvalue>. Intersecting with the previous enclosures therefore improves the last bound to
$$
\boxed{n-\epsilon\le\lambda_n\le n+\epsilon}.
$$
The identification with the same last <eigenvalue> is justified by separation from all the other integer-centred discs.