Let $X$ be a set. Let $w=(x_i)_{1\leq i\leq m}$ and $w'=(x'_j)_{1\leq j\leq n}$ for some $m,n\in\mathbb{N}$. The composition of $w$ and $w'$, denoted by $ww'$, is the family $(y_k)_{1\leq k\leq m+n}$ defined by
\begin{equation} y_k= \begin{cases} x_k & 1\leq k\leq m; \\ x'_{k-m} & m+1\leq k\leq m+n. \end{cases} \end{equation}
Let $w=(x_i)_{1\leq i\leq m}$ for some $m\in\mathbb{N}$. I want to show that $ew=w$, where $e=(u_i)_{1\leq i\leq 0}$ is the empty sequence. Applying the above definition, $ew$ is the sequence $(y_k)_{1\leq k\leq m+0}$ defined by
\begin{equation} y_k= \begin{cases} u_k & 1\leq k\leq 0; \\ x_{k-m} & 1\leq k\leq m. \end{cases} \end{equation}
But the expression $x_{k-m}$ makes no sense for $1\leq k\leq m$. What I am doing wrong? Am I applying the definition incorrectly?
Well, if you copy and paste your formula and replace $m$ by $0$, you get \begin{equation} y_k= \begin{cases} x_k & 1\leq k\leq 0; \\ x'_{k} & 1\leq k\leq n. \end{cases} \end{equation} which gives you $ew = w$ as required.