Can I substitute stuff inside $| \cdot|$ and retain inequalities?
Example: want to make
$$ \left|\frac{m}{n} - 1 \right| < \epsilon, \quad \epsilon > 0; \quad n,m \in \mathbb{N}$$
Could I say:
Take $$ n > \frac{m}{\epsilon+1} $$ so that $$ \left|\frac{m}{n} - 1 \right| < \left| \frac{m}{\frac{m}{\epsilon+1}} - 1 \right| = |\epsilon + 1 - 1| = \epsilon. $$
Now for this simple case this seems to work, since for a fixed and positive constant $1$ it's safely known that the distance of a larger positive quantity from that will lead to larger distance.
However, I don't think the $| \cdot |$ would always allow this kind of "positive real line" modifications?
Any background to this "property" or non-property of $|\cdot|$ or why not norms as well? What would one test for? That the operation is such that it's explainable either by homogeneity or triangle eq. (or possible C-S)?
In general, when do absolute values and norms retain inequalities?