 Solution

The Knill--Laflamme condition for code projector $P$ and errors $\{E_a\}$ isFor Pauli errors of weight at most $t$, the product $E_a^{†}{E}_b$ has weight at most $2t$. If $d\ge 2t+1$, such a product cannot be a nontrivial logical Pauli, because every such operator has weight at least the distance of a stabilizer code $d$. It is therefore either a stabilizer, acting as a scalar on the code, or it anticommutes with some stabilizer and maps the code to an orthogonal syndrome space. These are exactly the two possibilities required by the displayed condition, so every error on at most $t$ qubits is correctable.