For some reason, I have trouble getting absolute value right. This is of a great importance in the definition of the limit.
How do I solve the following inequality for $x$:
$$|x -a| < \epsilon$$
I know the solution is $a - \epsilon < x < a + \epsilon$, but I cannot provide the logical deductions. This is how I would start:
- $|x-a| < \epsilon$ $\iff$ $(x-a) < \epsilon$ and $(x-a) < -\epsilon$
- Simplify: $(x-a) < \epsilon$ and $-(x-a) > \epsilon$
This is where I think I go wrong because now I have a negative $x$.
Could someone show me the logical deductions step by step so I can see how it turns out?