## Question About Norms in Lp spaces and Chebyshev Norm

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

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

Best Regards, Dmitro.