i want to prove that if $K$ is an infinite field and $f\in K[X_1,X_2,\ldots,X_n]$ is a nonzero polynomial, then
There exists $c_1,c_2,\cdots,c_n\in K $ such that $f(c_1,c_2,\cdots,c_n)\neq 0$.
i wanted to prove this theorem by contradiction can you evaluate the following proof to see if it is correct or not?
proof: suppose that for all $c_1,c_2,\cdots,c_n\in K $ we have that $f(c_1,c_2,\cdots,c_n) = 0$. then since field must have Identity element we have $1\in K $ then for all $c_1 $ we have $f(c_1,1,\cdots,1) = 0 $ then it implies that for all
$f(c_1,1,\cdots,1) = $ $ \sum a_{i_1,\cdots, i_n} $ $ c_1^ {i_ {1}} $ $ \cdots $ $ c_n^ {i} $ = $ \sum a_{i_1,\cdots, i_n} $ $ c_1^ {i_ {1}} $ $ \cdots $ $ 1^ {i} $ = $ \sum a_{i_1,\cdots, i_n} $ $ c_1^ {i_ {1}} $ $ = 0 $
which implies that $ c_1 = 0 $ if we let that one coefficient be equal to 1 and all other coefficients be set to 0.because $c_1$ is a evaluation of a non zero polynomial. then for all of $c_1$ which belong to our field K we showed that $ c_1 = 0 $ hence we have a finite field hence we get a contradiction so our prior proposition must be true.
is this proof correct?