Kleene positive closure - help in proofing this claim

Problem Detail: 

I just started a course called 'Automata and Formal Languages'. I'm having difficulty in proofing\disproofing this equality.

$ (L_{1} \circ L_{2})^{+} = L_{1}^{+} \circ L_{2}^{+} $

Where:

$ L_{1} $, $L_{2}$ are Languages.

$\circ$ is the concatenation operation between two languages.

$+$ is the Kleene plus closure defined by $\bigcup _{i = 1}^{\infty }L^{i} $

I tried finding a counter example and also tried to formally proof but had no luck. Can someone please point me in the correct direction?

Asked By : elmekiesIsrael

Answered By : Yuval Filmus

Let's use $A$ for $L_1$ and $B$ for $L_2$. If you unpack both sides, you get $$ (AB)^+ = AB + ABAB + ABABAB + \cdots \\ A^+B^+ = AB + AAB + ABB + AABB + \cdots $$ Perhaps this can give you some ideas.

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Question Source : http://cs.stackexchange.com/questions/32509