Worked solution: Huang's signed hypercube adjacency matrix proof of the sensitivity conjecture (2019)

Fix any induced subgraph $H$ of $Q_n$ on a vertex set $S$ with $|S| \ge 2^{n-1}+1$, and let $A_H$ be the principal submatrix of $A_n$ obtained by keeping only the rows and columns indexed by $S$. By Lemma 2.2 (Step 3), $A_n$ has $\sqrt{n}$ as an eigenvalue of multiplicity $2^{n-1}$, which exceeds $2^n - |S|$; applying Cauchy's interlacing theorem (Step 2) with $i = 2^{n-1}$ therefore gives $\lambda_1(A_H) \ge \lambda_{2^{n-1}}(A_n) = \sqrt{n}$. Combining this with the row-sum lemma (Step 4) applied to $A_H$ yields $\Delta(H) \ge \sqrt{n}$ -- exactly Huang's Theorem 1.1.