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 trick is that 
$$|M-g(x)|=|g(x)-M|$$ because $$|M-g(x)|=|(-1)(g(x)-M)|=|-1|\cdot|g(x)-M|=|g(x)-M|$$
So that $$|(f(x)-g(x))-(L-M)|=|(f(x)-L)+(M-g(x))|\leq |f(x)-L|+|g(x)-M|$$
