Prove the sequence $\left\{ 2n \right\}_{n \in \mathbf{N}}$ is unbounded using the Archimedean Property and Russell’s Paradox.

Question

I know that in order of a sequence to be bounded, it should converge and in order for a sequence to converge, it should follow the following:

For all $\epsilon > 0$, there exists $M$ that belongs to $N$. for all $M \leq n$ that belongs to $N$, $| X_n - L | < \epsilon$.

What I have done so far in order to solve this specific sequence is that I negated the previous sentence in order to prove that the sequence diverges which means its unbounded.

So I said: there exist $\epsilon > 0$, for all $M$ that does not belong to $N$, there exists $M \leq n$ that belongs to $N$. Now, $|X_n - L | \geq \epsilon$ and I know that $\lim(2^n) = \infty$ which means that $L = \infty$.

However, I'm not sure what I'm supposed to do after that.