Can we make a sequence of real numbers such that polynomial of any degree with co-efficients of the sequence has all its roots real and distinct ? Does there exist a sequence of real numbers $(a_n)$ such that $\forall n \in \mathbb N$ , the polynomial $a_nx^n+a_{n-1}x^{n-1}+...+a_o$ has all $n$  real roots ? Can we make a sequence so that all the $n$ real roots are distinct ? 

Here is an inductive, semi-non-constructive approach: assume I have a polynomial $p(x) = a_0 + \dots + a_n x^n$, with $n$ real roots and which goes to infinity as $x$ goes to infinity. Let M be large enough that all the zeros of $p$ are contained in $[-M,M]$.
Now, pick $a_{n+1}$ positive but so small that $a_{n+1}x^{n+1}$ is essentially zero on $[-2M,2M]$. Then $p(x) + a_{n+1}x^{n+1}$ looks almost the same as $p(x)$ on the interval $[-M,M]$ (in particular it has all the old zeros, slightly perturbed), but it also has a new real root, somewhere to the right of $-2M$.
