Let $U$ be a subset of $\mathbb{R}^n$, and suppose that $U$ is not bounded. Construct a sequence of points $\{a_1, a_2, \ldots \}$ such that no subsequence converges to a point in $U$, then prove this claim about your sequence.
Let $U$ be a subset of $\mathbb{R}^n$ such that $U$ is not closed. Construct a sequence of points $\{a_1, a_2, \dots \}$ such that no subsequence converges to a point in $U$, then prove this claim about your sequence.
Could any one give some hints how to construct sequences in these two conditions?
Since $U$ is a subset of $\mathbb{R}^n$, I'm kind of confused how to construct such sequence in $\mathbb{R}^n$.
For the first sequence for each natural $m$ choose a point $a_m\in U$ such that $\|a_m\|>m$.
For the second sequence. Since the set $U$ is not closed, there exists a point $a\in\overline{U}\setminus U$. For each natural $m$ choose a point $a_m\in U$ such that $\|a_m-a\|<1/m$.