= Solution
Assume $0<\epsilon<1/4$, as implied by its stipulated smallness, and use the nontrivial matrix range $n\ge3$. The row radii are $\epsilon$ at one, $r=1/4+\epsilon$ at each interior index, and $1/4$ at $n$. Since $2r<1$, all discs are disjoint. The second theorem therefore gives one <eigenvalue> in each disc. Each such <eigenvalue> is real: a nonreal <eigenvalue> would have its distinct complex conjugate in the same disc, contradicting its count of one. Thus
$$
\boxed{|\lambda_1-1|\le\epsilon,\quad|\lambda_i-i|\le1/4+\epsilon\ (2\le i\le n-1),\quad|\lambda_n-n|\le1/4}.
$$
The eigenvalues are simple as well. An alternative reality argument is that a positive diagonal similarity makes this tridiagonal matrix symmetric, since its opposite off-diagonal entries have positive product.
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