Sequence with an infinite amount of limit points 
Find a sequence which has an infinite amount of limit points.

I was thinking about using the bijective pairing function $\langle\cdot,\cdot\rangle:\Bbb N\times\Bbb N\to\Bbb N,\langle x,y\rangle=\binom{x+y+1}{2}+x$ with $\pi_1(\langle x,y\rangle)=x$ and $\pi_2(\langle x,y\rangle)=y$ to describe the sequence
$$a_n=\frac{\pi_1(n)}{\pi_2(n)}.$$
In this case all numbers in $\Bbb Q$ are part of $a_n$, thus all numbers from $\Bbb R$ should be limit points due to $\Bbb Q$ being dense in $\Bbb R$.
Is this solution right or do you have an even easier solution?

Let $X_n$ be the largest prime factor of $n$. All primes numbers will be limit points of this sequence. As there are an infinite number of prime numbers, we get the desired sequence.
