Prove that for $f,g\in F[x]$, where $F$ is an infinite field, if $f(a)=g(a)$ for infinitely many elements $a\in F$, then $f=g$ 
Prove that for $f,g\in  F[x]$, where $F$ is an infinite field, if $f(a)=g(a)$ for infinitely many elements $a\in F$, then $f=g$.

I'm not entirely sure how to tackle the "infinitely many elements of a of field F" portion of this proof.  How would you even account for all those a's?

Actually, the number of zeros of a non-zero polynomial is less then or equal to its degree.
This is a general theorem which is true for any polynomial over any field.
(Note that the number can indeed be  less then the degree, for example $p(x)=x^2+1$ has no zeros at all in $\mathbb{R})$.
Now, assuming the above result, reaching a contradiction is easy: assume $f(x)\neq g(x)$ are two different polynomials. Then their difference $f(x)- g(x)$ is a non-zero polynomial. 
Hence, its number of roots is bounded by its degree and in particular, is finite: $\Rightarrow$ contradiction.
