How can I prove this using the hint? Also I can only use Bolzano's thm and basic definition of limit and boundedness or any basic topology.

5. Let X : [a,a]. Let p : X > R be a polynomial function. Assume that the function satises p(0) = 0. Show that, for every 5 > 0 there exists a natural number M E N such that |p(93)| S 6 + M562, for all :0 E X. (Hint: By way of contradiction, suppose that it is not the case. Show that there exists 50 > 0 and a sequence (53\") in X such that lp(wn)| > 60 + 71- 36$- Find a convergent subsequence such that $11k > w*. Show that m* = 0 and |p(:c*)| 2 50 > 0.)