Consider the following statement $S_a$, with some parameter $a > 1$ :

"If $P(X)$ is a real polynomial such that for every $i\in \{0;1;\ldots;N\}$, $|P(a^i)|< 1$ and $|P(a^0)-P(a^1)|>1$, then the degree of $P$ is $\Omega(N)$."

It happens to be true for all $a>1$, and the lower bound on the degree of $P$ can be more or less made uniform in $a$, but my question is the following: if you know $S_a$ to be true for some given $a>1$, is there a direct way to deduce that it is true for all of them? Intuitively, it is blatantly obvious that it should be the case, but I fail to find a correct argument.