Norm of a functional on $C[a,b]$
$\varphi$ is a continuous function on a compact set, so it's sup is attained at some point $y \in [a,b]$. Take a sequence of functions $f_n$ such that $f_n \to \delta_y$, the Dirac mass at $y$. If you are unaware, the Dirac mass satisfies $\int \delta_y\varphi(x) dx = \varphi(y)$, and since $y$ is the point the extrema is attained at, this is your desired sequence of functions.