absolute value inequality understanding or proof

Question

I'm looking at this inequality conclusion in a proof I'm reading and I can't seem to understand it. I was wondering if somebody can explain it to me or perhaps give me, or point me to a proof.

$\frac{p_m}{m} - \frac{p_n}{n} < \frac{1}{n}$ and $\frac{p_n}{n} - \frac{p_m}{m} < \frac{1}{m}$,

Therefore

$\lvert \frac{p_n}{n} - \frac{p_m}{m} \rvert < \frac{1}{n}+\frac{1}{m}$

Edit: $n, m \in \mathbb N$

Answer 1

I'm going to assume that $m, n \geq 0$, otherwise the statement is not true.

Essentially, what your inequalities say is that if $x < a$ and $-x < b$ (with $a, b, \geq 0$), then $|x| < a + b$. This can be easily proven by cases.

Either $|x| = x$ or $|x| = -x$. In the first case we have $|x| < a$ and in the second case we have $|x| < b$. Thus $|x| < \text{max}(a, b) \leq a + b$ since $a, b \geq 0$.

Answer 2

Negate the first inequality and combine it with the second to obtain $$-\frac{1}{n} < \frac{p_n}{n} - \frac{p_m}{m} < \frac{1}{m}$$ Hence, $$\left|\frac{p_n}{n} - \frac{p_m}{m}\right| < \max\left\{\frac{1}{m}, \frac{1}{n}\right\} < \frac{1}{m} + \frac{1}{n}$$