How does this work? $| [f(x)−g(x)] − (L−M) | \leq | f(x)−L | + | g(x)−M |$ My teacher was showing my class a proof for the limit difference rule using the epsilon-delta definition, and nearing the end, he showed us this: 
|[f(x)−g(x)] − (L−M)| ≤ |f(x)−L| + |g(x)−M| . I know what the triangular inequality is, and how it works, but on the left, he has a subtraction not a addition. Are there steps he didn't show, or did he just make a mistake? 

The triangle inequality gives
$$\begin{align}\left|[f(x)-g(x)]-(L-M)\right|&=\left|[f(x)-L]-[g(x)-M]\right|
\\&\leq \left|f(x)-L\right|+\left|-[g(x)-M]\right|
\\&=\left|f(x)-L\right|+\left|g(x)-M\right|\end{align}$$
