[Solved]: 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 $Xsubseteq mathbb{N}$ be any undecidable set containing $1$ and every even number. For example, take your favourite undecidable set $S$ and let $$X = {0, 2, 4, dots} cup {1} cup {2i+1mid iin S},.$$ The language $mathcal{L} = {a^imid iin X}$ is undecidable, so it certainly isn’t regular. But $$mathcal{L}cdotmathcal{L} = {a^{i+j}mid i,jin X} = {a^imid iinmathbb{N}},,$$ is regular.