change the definition of sequence limits

Question

For a sequence $_{k\ge p}$ in $\mathbb R$ and for $ a \in \mathbb R$ we say a is a $limiting \ value$ of $_{k\ge p}$ when $\forall \epsilon \gt0 ,\forall m \in \mathbb Z ,\exists k \in \mathbb Z_{\ge p}\space (k \ge m \ and \ |x_{k} - a|\leq \epsilon)$. We denote the set of limiting values of $_{k\ge p}$ by $Lim(_{k\ge p})$

1. Prove or disprove , for every sequence $_{k\ge p}$, we have $\lim_{k \to \infty}x_{k} = a \Rightarrow Lim(_{k\ge p}) = \{a\}$
2. Prove or disprove , for every sequence $_{k\ge p}$, we have $Lim(_{k\ge p}) = \{a\} \Rightarrow\lim_{k \to \infty}x_{k} = a$
3. Determine whether or not there exists a sequence $_{k\ge p}$ in $\mathbb R$ with $Lim(_{k\ge p}) = \mathbb R$

I think both (1) and (2) are True and I proved (1). But I really have no idea how to prove or disprove (2) and (3)... anyone can give some hints? Thanks !

Answer 1

2 is false. Consider $x_k=(1+(-1)^k)k$.

For 3, recall that $\Bbb Q$ is a countable set.