In this paper we consider sequences of best approximation. We first examine the rho best approximation function and its applications, through an example in approximation theory and two new examples in calculating n-widths. We then further discuss approximation theory by examining a modern proof of Weierstrass's Theorem using Dirac sequences, and providing a new proof of Chebyshev's Equioscillation Theorem, inspired by the de La Vallee Poussin Theorem. Finally, we examine the limits of approximation theorem by looking at Bernstein Lethargy theorem, and a modern generalization to infinite-dimensional subspaces. We all note that smooth functions are bounded by Jackson's Inequalities, but see a newer proof that a single non-differentiable point can make functions again susceptible to lethargic rates of convergence.