Proving that non-regular languages are closed under concatenation
You can't prove it because it isn't true: the class of non-regular languages isn't closed under concatenation.

Let $X\subseteq \mathbb{N}$ be any undecidable set containing $1$ and every even number.  For example, take your favourite undecidable set&nbsp;$S$ and let $$X = \{0, 2, 4, \dots\} \cup \{1\} \cup \{2i+1\mid i\in S\}\,.$$ The language $\mathcal{L} = \{a^i\mid i\in X\}$ is undecidable, so it certainly isn't regular.  But $$\mathcal{L}\cdot\mathcal{L} = \{a^{i+j}\mid i,j\in X\} = \{a^i\mid i\in\mathbb{N}\}\,,$$
is regular.