This is for young people doing CS and are wondering how to use the Pumping Lemma to verify if a formal language is regular or not :-). So let us have a theorem about the balanced parenthesis language :
Theorem:
$L_{()} = \{ (^n \varphi )^n | n \geq 1 \wedge \varphi \neq \Lambda \}$ is not a regular language.
Method:
Some text books use $\Lambda$ to denote the empty string so here $\varphi$ is none empty. The method of proof is by contraction (Reduction ad Absurdium – RAA). So we assume the language is regular and hence can be pumped but show it to be otherwise, or not true by arriving at a contradiction. By the Pumping Lemma (PLem), $\exists m$ so that $m$ can be used to split $w \in L_{()}$ into parts. We actually use this $m$ to form $w$. We can do this because the Lemma says this $m$ is valid for any $w, |w| \geq m$.
Proof:
Assume $L_{()}$ to be regular. Then by PLem,  there exists an $m$ such that for $w \in L_{()}$, $|w| \geq m$. Consider now the $w$ formed by setting $n = m$. Then we have $w = (^m \varphi )^m$. Further we know that $|w| = 2m+1 > m$ satisfying PLem premise, so it can be applied. As per PLem,  we can break $w$ into $x, y, z$ components. As per PLem also $|xy| \leq m$. Looking at the form of our $w$ this implies that $xy$ must be composed of all left parenthesis, i.e  $xy = (^m$, then $y=(^p$ for some $p < m$.  As per PLem, we can pump $y$  for any $k \geq 0$.  So let $k =0$ then the new $w = (^{m-p} \varphi )^m$. But this implies that $|(^{m - p}| = m - p \neq m =|)^m|$, i.e. parentheses are not balanced. But this means $w \not \in L_{()}$. Contradiction. At this point we have found a $k$ where the decomposition results in the string outside of $L_{()}$and we can stop.

Anyway consider now also $y$ pumped upwards to $k$. Then we have the new $w = (^{m-p}(^{pk} \varphi )^m$. Looking at the left of $\varphi$ we have $|(^{m-p}(^{pk}|= m - p + pk \neq m= |)^m|$. Again the parentheses do not balance out. Thus $w \not \in L_{()}$. Contradiction.

Q.E.D.

Feedback is appreciated – let me know if it has helped/not helped. Thanks.