Originally Posted by

**owq** Ok, I see that when $\displaystyle |x| < \delta + 2$ it contains all the values from $\displaystyle 0<|x+2| < \delta$.

But aren't we trying to fulfill the inequality $\displaystyle 0<|x+2| < \delta$? If $\displaystyle |x| < \delta + 2$ then let's say that when $\displaystyle x = \delta + 1.9$, it will fulfill $\displaystyle |x| < \delta + 2$ but not $\displaystyle 0<|x+2| < \delta$ which was our original intention.

Similarly for $\displaystyle \left|{x + 2} \right| < 1\, \Rightarrow \,\left| x \right| < 3\$, let's say $\displaystyle x = 2$, then it would fulfill $\displaystyle |x|< 3$ but not $\displaystyle |x + 2| < 1$.

Did I get my concepts wrong somewhere?