= Solution
Write $q=1/4+\epsilon$ and take $0<\epsilon<1/4$, as intended by the small-parameter assumption. The row discs have centers $1,\ldots,n$ and radii $\epsilon,q,\ldots,q,1/4$. They are pairwise disjoint because the sum of neighboring radii is below one. The second <Gershgorin circle theorem> places one <eigenvalue> in each disc. Since $A$ is real, nonreal <eigenvalues> occur in conjugate pairs; a disc centered on the real axis would have to contain both members, contradicting its count of one. Thus all <eigenvalues> are real and
$$
\boxed{|\lambda_1-1|\le\epsilon,\quad
|\lambda_i-i|\le\tfrac14+\epsilon\ (2\le i<n),\quad
|\lambda_n-n|\le\tfrac14.}
$$
Alternatively, a positive diagonal change of <basis> symmetrizes the tridiagonal <matrix>, with transformed off-diagonal entries $\sqrt\epsilon/2$, also proving reality.
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