How does this work? $| [f(x)−g(x)] − (L−M) | \leq | f(x)−L | + | g(x)−M |$

Question

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?

Answer 1

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

Answer 2

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 $$|(f(x)-g(x))-(L-M)|=|(f(x)-L)+(M-g(x))|\leq |f(x)-L|+|g(x)-M|$$