Question About Norms in Lp spaces and Chebyshev Norm

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

We assume that $\displaystyle A$ is a finite dimensional linear subspace of the normed linear space $\displaystyle C[0,1]$ of continuous functions on the interval $\displaystyle [0,1]$. We let $\displaystyle f$ be in $\displaystyle C[0,1]$ and define $\displaystyle a(p)$ to be the best approximation from $\displaystyle A$ to $\displaystyle f$ as defined by the $\displaystyle p$ norm for functions. We define $\displaystyle a*$ to be the best approximation from $\displaystyle A$ to $\displaystyle f$ as defined by the Chebyshev norm. Does the sequence $\displaystyle {a(1),a(2),...} $ of best approximations to $\displaystyle f$ in $\displaystyle Lp$ converge to $\displaystyle a*$ as $\displaystyle p$ approaches infinity?

Best Regards, Dmitro.