I am reading the following writing by Keith Conrad: https://kconrad.math.uconn.edu/blurbs/ugradnumthy/crt.pdf
I don't understand the remark 4.15. Where is the error in the following proof:
1) For every $i$ find an integer $a_i$ such that $f(a_i)\not\equiv 0 \pmod{p_i^2}$
2) Use the Chinese remainder theorem to find integer $a$ such that $\forall i \quad a\equiv a_i \pmod{p_i^2}$
3) $\forall i \quad f(a) \equiv f(a_i) \pmod{p_i^2} $, because $(x-y) |(f(x)-f(y))$ if $f$ is polynomial with integer coefficients.
4) $\forall i \quad f(a) \equiv f(a_i) \not\equiv 0 \pmod{p_i^2} $, contradiction.