As a counterpart to the best approximation in normed linear spaces, the best simultaneous approximation was introduced. In this paper, we shall consider relation between simultaneous proximinality $W$ in $X$ and $L^p(S,W)$ in $L^p(S,X)$ for $1leq pleqinfty$. Also, we consider the relation between w-simultaneous proximinality $W$ in $X$ and $L^p(S,W)$ in $L^p(S,X)$ for $1leq pleqinfty$.