Introduction
Let $f \in C(\mathbb{R}^n)$, supp $f \subset \lbrace x : |x| <1 \rbrace$, and $\int_\mathbb{R} f(x) dx = 1$.
Consider the sequence $f_k(x) = k^n f(kx)$, $k \in \mathbb{R}$.
Show that $f_k(x) \rightarrow \delta(x)$ as $k \rightarrow \infty$ in $\mathit{D'(\mathbb{R}^n)}$.
My main problem here is with the integral. $f_k(x)$ takes $x \in \mathbb{R}^n$ but in the integral in the first line, we only integrate over $\mathbb{R}$.
So when I start the proof I have $\int_\mathbb{R^n}k^nf(kx)\varphi(x) dx$ for $\varphi \in \mathit{D}(\mathbb{R}^n)$.
I was thinking of maybe having n integrals, so something like $\int_\mathbb{R} kf(kx) \int_\mathbb{R} kf(kx) \dots \int_\mathbb{R} kf(kx)\varphi(x) dx_1\dots dx_n$ but I don't think this is right since we have $dx$ and not $dx_i$ in the integral given at the beginning.
I'm sure I'm making a silly mistake somewhere, any hints would be appreciated!