Let $F = \{f : [a, b] \to R,\ f \in C^1([a, b])\ \text{and}\ f(a) = 0\}$ with fourier coefficients as the inner product.
While $f_n(x) = \int_{0}^{x} g_n(y)dy$ is a cauchy sequence of functions on [-1, 1] where $g_n$ converges to sgn in the $L^2$ norm, and f_n(x) converges to the absolute value function for $x > 1/n$ in F's norm, I would like an example of a sequence of differentiable functions with supp(a,b] that converges to a function that is non zero on [a,b], which makes the solution much cleaner.
I tried a sequence of functions that have supp(0,1] and converge to a function with support on the compact interval [0,1], something along the lines of $(x - 1/n)^2$, but unfortunately $x^2$ of 0 is 0. I thought this would make it much easier, but I can't think of such a sequence.
Thanks