Problem 191 (difficulty: 3/10)

Let \(\displaystyle a_n\) be a sequence and \(\displaystyle a\) be a number. What are the implications among the following statements?

a) \(\displaystyle \forall \varepsilon>0\) \(\displaystyle \exists N\) \(\displaystyle \forall n\geq N\) \(\displaystyle |a_n-a|<\varepsilon\).

b) \(\displaystyle \forall \varepsilon>0\) \(\displaystyle \exists N\) \(\displaystyle \forall n\geq N\) \(\displaystyle |a_n-a|\geq\varepsilon\).

c) \(\displaystyle \exists \varepsilon>0\) \(\displaystyle \forall N\) \(\displaystyle \forall n\geq N\) \(\displaystyle |a_n-a|<\varepsilon\).

d) \(\displaystyle \forall \varepsilon>0\) \(\displaystyle \forall N\) \(\displaystyle \forall n\geq N\) \(\displaystyle |a_n-a|<\varepsilon\).

e) \(\displaystyle \exists \varepsilon'>0\) \(\displaystyle \forall 0<\varepsilon<\varepsilon'\) \(\displaystyle \exists N\) \(\displaystyle \forall n\geq N\) \(\displaystyle |a_n-a|<\varepsilon\).

Give me another random problem!