Let there be a finite character set $\Sigma$, as in computer science convention. $\Sigma^{*}$ is defined as in Kleene star notation (https://en.wikipedia.org/wiki/Kleene_star) with $\Sigma^{+}$ defined as Kleene plus of $\Sigma$.
Now proving $\Sigma^{+} = \Sigma^{*}\Sigma = \{ab | a \in \Sigma^{*} \wedge b \in \Sigma \}$. I know that I can prove by saying because all strings in $\Sigma^{*}$ contain finite characters, adding another character to right does not change finiteness of character, and because $\Sigma^{*}$ contains an empty character, $\Sigma^{+} = \Sigma^{*}\Sigma$. However, I cannot find a formal way of proving this.
Can anyone help here?
If you use the definitions in the Wikipedia article to which you linked, you have
$$\begin{align*} \Sigma^+&=\bigcup_{k\ge 1}\Sigma_k\\ &=\bigcup_{k\ge 0}\Sigma_{k+1}\\ &=\bigcup_{k\ge 0}\left(\Sigma_k\Sigma\right)\\ &=\left(\bigcup_{k\ge 0}\Sigma_k\right)\Sigma\\ &=\Sigma^*\Sigma\;. \end{align*}$$