Why is it stronger statement for all positive integers $x$, but $x$ is real number in the original theorem I am reading book A Walk Through Combinatorics, but I don't understand why the writer says "prove an even stronger statement".

Corollary 5.10. For all real numbers $x$, and all non-negative integers $n$
$$x^n=\sum_{k=0}^nS(n,k)(x)_k$$
Proof: Both sides are polynomials of $x$ of degree $n$. So if we can show that they agree for more than $n$ values of $x$, we will be done. We will prove an even stronger statement, namely that the two sides agree for all positive integers $x$. 
So let $x$ be a positive integer. Then...

Where did I get it wrong?
Update:
Another question I don't understand is how to derive "So if we can show that they agree for more than $n$ values of $x$, we will be done." from "Both sides are polynomials of $x$ of degree $n$". Why?
 A: Hint: Assume two polynomials of degree $n$ agree at more than $n$ points.  Obviously the difference of the polynomials is also a polynomial of degree $k\le n$. By our assumption the polynomial has at least $n+1$ roots, but by fundamental theorem of algebra a non-zero polynomial of degree $k$ cannot have more than $k$ roots. It follows that the difference of polynomials is identically zero.
A: If two functions agree for all positive integers $x$, then obviously they agree for more than $n$ real values of $x$, so in that sense they are proving a stronger statement. So it sounds like they're announcing a logical outline for the proof:

*

*The two functions agree for all positive integers $x$.

*Therefore they agree for more than $n$ real values of $x$.

*The two functions are polynomials of degree $n$.

*If two polynomials of degree $n$ agree for more than $n$ real values, they are identical.

*Therefore the two functions are identical.

Note, step 4. is a general assertion about polynomials, included here to address the question in the OP's update.
