Korovin Theorem: Let $K \subset \mathbb{R}$ be a compact set. Let $(L_n)$ be a sequence of $LPO$ on $C(K,\mathbb{R},||.||_{\infty})$. If $L_n$ converge on the functions $x \to 1, x \to x, x \to x^2$, then $L_n$ converges on all $C(K,\mathbb{R},||.||_{\infty})$.
The proof (at least the ones that I know) in one dimensional is not difficult, a game of $\epsilon, \delta$. However, studying Bernstein polynomials $B_n$ in order to apply Korovi's Theorem one studies the convergence of the operator on the three functions of the Theorem. On $x \to 1$ and $x \to x$, $B_n$ is exact, but $B_n[x^2](y) = y^2-\frac{1}{n}y^2+\frac{1}{n}y$ so it converges to $x \to x^2$ with $n$. One observation in my notes is the following: "We couldn't expect that the error was zero even this time, otherwise in Korovin's theorem..."
But I don't know how to see some contradiction in Korovin's proof. However from the proof emerges that if we have exactness on $x \to 1, x \to x, x \to x^2$, then for every $n \geq 1, \epsilon > 0$ holds $||L_n(f)-f||_{\infty,K} < \epsilon$. There is a contradiction in here?
Any help or clarification would be appreciated