I take it that what you have in "The attempt at a solution",
[tex]x^n\le a_0+ a_1x+ a_2x^2+ \cdot\cdot\cdot+ a_{n-1}x^{n-1}[/tex]
is actually what you want to prove. I cannot open the first attachment. But that is certainly NOT true for general x, [itex]a_0[/itex], [itex]a_1[/itex], ... So what are you really trying to do?