Proof by "infinite induction" Prove that $\sum_{i=1}^{n} i^3 = \left( \frac{n(n+1)}{2} \right)^2$.

We can check this is true for n=0,1,2,3,4. Since the right side is a polynomial of degree 4, and the left side is a sum of monomials whose degree is &lt;= 4, then if both polynomials coincide for five points, they must be the same.

My question: is this proof rigorous? I'm concerned about the left side sum.
No, your proof is not rigorous. It is actually wrong. The point is that you don't know a priori that the left-hand-side is a polynomial of degree $4$ on $n$. 

As an example of your "reasoning", consider the equality
$$
\sum_{i=1}^n i = 2n-1.
$$
This is of course false (the actual formula on the right should be $n(n+1)/2$. But, for $n=1$ and $n=2$, both sides agree ($1$ and $3$ respectively), and the right hand side is a polynomial of degree $1$, so two points would suffice to determine it, if we were allowed to reason as you did.
Yes this is rigorous because the a basic fact about polynomials is that the discrete integral of a degree $d$ polynomial is a degree $d+1$ polynomial.