I'm stuck on 7. (c) (see problem below) proving that this polynomial has no more than $m$ roots (so far I have that it has $\geq m$ roots). I am trying to proceed by contradiction, assuming that there is such a coefficient vector $d \in \mathbb{R}^n$ contained in every neighbourhood about $c$ (our coefficient vector given in the problem) such that $$x^n + d_{n-1}x^{n-1} + ... + d_0$$ has $> m$ roots. My gut tells me to take a sequence of these $d$s that converge to $c$ and use the continuity of functions guaranteed by the Implicit function theorem but this doesn't seem to lead me anywhere since these functions are only guaranteed on neighbourhoods about $d$ that need not contain $c$. Can someone give me a hint? What should I be focusing on?
