I would really appreciate it if you could give me some advice on the following exercise in Rudin.
Put $K=[-1,1]$; define $\mathcal{D}_K$ as the set of all smooth functionals supported in $K$. Suppose $\{ f_n \}$ is a sequence of Lebesgue integrable functions such that $$\Lambda \phi = \lim_{n\to \infty} \int_{-1}^1 f_n(t)\phi(t) dt$$ exists for every $\phi \in \mathcal{D}_K$. Show that $\Lambda$ is a continuous linear functional on $\mathcal{D}_K$. Show that there is a positive integer $p$ and a number $M<\infty$ such that $$| \int_{-1}^1 f_n(t)\phi(t) dt| \le M \|D^p \phi \|_\infty$$ for all $n$. Construct an example where it can be done with $p=2$ but not with $p=1$.
I proved that $\Lambda$ is a continuous linear functional, thus completing the first part. However I cannot see how to approach the second part. I tried to use the uniform boundedness of linear functionals $\Lambda_n:\phi \mapsto \int_{-1}^1 f_n(t)\phi(t)dt $ but failed to solve it.