I have these stated inequalities, true for all $i$.
$$\begin{align} |P_{i}+q_{i}|\leq \epsilon \tag{1} \end{align}$$
with $\epsilon > 0$ and I know that $$\begin{align} |P_{i}-P_{j}|\leq \delta \tag{2} \end{align}$$
My problem is that by using (1) I want to show that (2) can be re-written in
$$|q_j-q_i|-2\epsilon \leq \delta$$
with $\delta > 0$. But I can't... I'm always blocked at the first step.
$$\begin{align} |P_{i}-P_{j}|&\leq \delta \\ ||P_{i}+q_i-q_i|-|P_{j}+q_j-q_j||&\leq \delta \\ ?? |||P_{i}+q_i|-|q_i||-||P_{j}+q_j|-|q_j|||&\leq \delta \quad??\\ \end{align}$$
or
$$\begin{align} |P_{i}-P_{j}|&\leq \delta \\ |(P_{i}+q_i)-(P_{j}+q_j)+(q_j-q_i)|&\leq \delta \\ \end{align}$$
but then what ? Since $-\epsilon\leq(P_{i}+q_i) \leq \epsilon$, can I simply say that the expression here $(P_{i}+q_i)-(P_{j}+q_j)$ is at maximum $2\epsilon$ and minimum $-2\epsilon$ and that that implies that the result is evident ?
I'm not able to go further... I only used these triangular inequalities $$\begin{align} |x+y|&\leq |x|+|y| \\ |x-y|&\geq ||x|-|y|| \end{align}$$ to get to my result. Should I use some other things because I don't see anything? Is the fact that both $-1\leq P_i\leq 1$ and $-1\leq q_i \leq 1$ an important fact I could use ? Or maybe it's impossible to prove it?
