Question About Norms in Lp spaces and Chebyshev Norm

Here is my question; I appreciate any and all help and hints.

We assume that is a finite dimensional linear subspace of the normed linear space of continuous functions on the interval . We let be in and define to be the best approximation from to as defined by the norm for functions. We define to be the best approximation from to as defined by the Chebyshev norm. Does the sequence of best approximations to in converge to as [LaTeX ERROR:
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approaches infinity?

Best Regards, Dmitro.