Flirting Sequences (Real Analysis) I need help with a homework problem and am pretty sure my real analysis teacher made the following definition up:

In a metric space, a sequence $\{P_n\}_n$ $ \ $ flirts with $p$ iff for each $\epsilon &gt; 0$, there is a $n \in \mathbb{N}$ and $m &gt; n$ such that $0 &lt; d(p_n,p) &lt; \epsilon$ and $d(p_n,p) &lt; d(p_m,p)$. The sequence $\{P_n\}_n$ $ \ $ is a flirting sequence if there is a point $p$ such that $\{P_n\}$ flirts with $p$. 

Give an example of a sequence that flirts with $1$.

I am trying to $(1)$ understand the concept of "flirting" and $(2)$ figure out an example of a sequence that flirts with $1$. Actually, I would be happy to see an example of a any sequence that flirts with something.

I translated the definition like this:

In a metric space, the sequence $\{P_n\}$ flirts with $p$ iff
$$
\forall \epsilon &gt; 0 \quad \exists(m,n \in \mathbb{N}, m &gt;n): 0 &lt; d(p_n,p) &lt; \epsilon \quad  \text{and} \quad  d(p_n,p) &lt; d(p_m,p).
$$

I have concluded that in $\mathbb{R}$ with the usual metric, the sequences {$2-\frac1n$} and {$1-\frac1n$} do not flirt with $1$. 

Can anyone help me understand this concept and/or provide an example of any sequence that flirts to some point?
Consider the following real-valued sequence $(p_n)_n$ which flirts with $0$:
$$p_n = \begin{cases} 1 &amp; \text{ if } n \text{ is even}\\
\frac{1}{n} &amp;\text{ if } n \text{ is odd}
\end{cases}$$ can you see why this is the case?



Note that


a sequence can flirt with $p$ but not converge to $p$: as in the above  (it will have a subsequence converging to $p$, though)
a sequence can flirt with $p$ and converge to $p$: 
$$p_n = \begin{cases} \frac{1}{2^n} &amp; \text{ if } n \text{ is even}\\
\frac{1}{n} &amp;\text{ if } n \text{ is odd}
\end{cases}$$
Remember that a easy example of a flirting sequence is $~-1, ~1/2,~ -1/3,~ 1/4, ~-1/5, ~1/6,~$ etc.

Mainly if it is even it is positive, if it is odd it is negative, just like $~(-1)^n~.~$

I think the formula is  $-1/n,~~ +1/(n+1)$

Even better, $-1/(2*(n)),~~ +1/(2*n+1)$.