I'm trying to solve this problem about distributions:
"Let $f\in\mathcal{D}(\mathbb{R})$ a test function with $\mathrm{supp}(f)\subseteq(-1,1)$ and $\||f|\|_1=1$. For each $k\in\mathbb{N}$, let $f_k$ to be defined by $f_k(x)=\int_{1/k}^kf(k(x-t))\,\mathrm{d}t$. Denote by $\Lambda_k$ the usual functional on $\mathcal{D}(\mathbb{R})$ given by $$\langle \Lambda_k,\varphi\rangle=\int_\mathbb{R} \varphi(x)f_k(x)\,\mathrm{d}x.$$ Show that $\Lambda_k\in\mathcal{D}^\prime(\mathbb{R})$ and $\Lambda_k\stackrel{\mathcal{D}^\prime(\mathbb{R})}{\longrightarrow}H$, where $H$ is the Heaviside function."
My attempt: The expression $\int_\mathbb{R}\int_{1/k}^k\varphi(x)f(k(x-t))\mathrm{d}t\mathrm{d}x$ remember us the use of approximate identities, so I think that we could use the ideas behind that approach. However, we found the with the problem that appears $"-t"$ and it should be $-x$; moreover $f$ is not non negative in general and also the inf and sup limits of the inner integral cause me a conflict.
Do you have some idea?