The relevant inequality for this problem is [itex]f_n(x)f(x) \leq f_nf_{\infty}[/itex] for all [itex]x \in [0,1][/itex]. But honestly, based on your threads in the homework help forums, you really lack the mathematical maturity to be dealing with convergence in function spaces. It would be worthwhile for you to go back and review basic convergence in [itex]\mathbb{R}[/itex].
