Mantel's theorem says that a triangle-free graph has at most $\frac{n^2}{4}$ edges.
Many proofs exist, including the following proof by induction: We consider an arbitrary edge $(x,y)$ and remove both vertices $x$ and $y$. We are left with a graph on at most $\frac{(n-2)^2}4$ edges, and we removed at most $n-1$ edges since $d(x)+d(y)\leq n$. So $m \leq \frac{(n-2)^2}4 + (n-1)=\frac{n^2}4$.
In this proof we removed two vertices from our graph, since the assumption on triangle-freeness constraints their sum of degrees. Alas, this assumption does not constrain the degree of a single vertex. How can we prove Mantel's theorem by induction removing only a single vertex?
The lecturer told us to "think of a fix" that would allow us to use induction in this way, but the original method cannot work since $\frac{n^2}4-\frac{(n-1)^2}4=\frac{2n-1}{4}$ is not a bound on the number of edges removed.