Question: Use the definition of absolute value and systems of inequalities to prove that for any real numbers x and c, and any positive real number δ, the given statement is true:
$|x−c|<δ⇔x∈(c−δ,c+δ)$
I am very confused on how to do this question as I am unable to get what the question is asking me. This question has been listed in one of my homework questions so I am very lost. If anyone can help me out, I'd appreciate it!
As with any bi-implication, it is asking you to prove two things:
That if $|x-c|\lt \delta$, then $x$ must be in the interval $(c-\delta,c+\delta)$; and
That if $x$ is in the interval $(c-\delta,c+\delta)$, then it must be the case that $|x-c|\lt\delta$.
It also instructs you to prove these facts using “the definition of absolute value and systems of inequalities”. Exactly what that means for your course is for you to know and not for us to guess; there are many different ways in which the basic properties can be set forth (usually any small number of assertions can be used to establish all the others), so exactly which one is being used in your course will be key in how you prove this.