[Exercise about distribution theory; definition of a distribution and examples:]
Let $\Omega \subset \mathbb R^{n}$,$f \in L^{1}_{loc}(\Omega)$, $ \beta \in \mathbb{N}^{n}$ multi-index. Consider the linear functional $u$ defined by $u(\varphi)=\int_{\Omega} f(x) \partial^{\beta} \varphi(x) dx$, $\forall \varphi \in C^{\infty}_{0}(\Omega)$. Prove $u$ is a distribution in $\Omega$, having finite order $\lvert\beta\rvert$.
[My attempt: ]
It is not difficult to prove $u$ is a distribution. Indeed, let $K \subset \Omega$ a compact subset, and $\varphi \in C^{\infty}_{0}(\Omega)$:
$$ \lvert u(\varphi) \rvert \le \biggl\lvert \int_{K} f(x) \partial^{\beta} \varphi(x) dx \biggr\rvert \le \int_{K} \lvert f(x) \rvert \lvert\partial^{\beta} \varphi(x) \rvert dx \le \sup_{K} \lvert \partial^{\beta} \varphi(x) \rvert \int_{K} \lvert f(x) \rvert dx =$$
$$ = \sup_{K} \lvert \partial^{\beta} \varphi(x) \rvert \lvert\lvert f \rvert\rvert_{L^{1}(K)} \le \lvert\lvert f \rvert\rvert_{L^{1}(K)} \sum_{\lvert \alpha \rvert \le \lvert \beta \rvert} \sup_{K} \lvert \partial^{\alpha} \varphi(x) \rvert $$
Thus, $u$ is a distribution of finite order. Such order is $\ge \lvert \beta \rvert$: indeed, I have the same case as before for every $K_1 \subset \Omega$ compact, $\lvert u(\varphi) \rvert \le \lvert\lvert f \rvert\rvert_{L^{1}(K_1)} \sum_{\lvert \alpha \rvert \le \lvert \beta \rvert} \sup_{K_1} \lvert \partial^{\alpha} \varphi(x) \rvert$.
However, it is not trivial to me to prove order $\le \lvert \beta \rvert$. Roughly speaking, if order is $< \lvert \beta \rvert$, I don't have information about $\sup_K \lvert \partial ^{\beta} \varphi \rvert$ and the inequality involving $\sum \sup_K \lvert \partial ^{\alpha} \varphi \rvert$ may lead to a contradiction. The problem is I don't know how to exactly prove this. Is my guess correct? Do you have any suggestions to go further that way? Do you think there is a better strategy?