Is it possible to prove that the limit, as n goes to infinity, of the sequence $a_n = 3^n/7^n$ is $0$? Intuitively this seems to be true, but is there a way to actually prove it instead of just saying "the denominator gets bigger faster"? I'm not sure that the sequence can be modeled by a function, but even in doing that and using l'Hopital's rule I didn't get 0.  

The argument relies on the completeness property of $\mathbb{R}$, or at least the Archimedean property. A proof in $\mathbb{R}$ using the standard completeness axiom:
Let $a_n=\left(\frac37\right)^n$. This sequence is bounded below by $0$, so it has a greatest lower bound $L$. It's a decreasing sequence, so $L$ is its limit.  Now, the sequence $b_n=\frac37a_n$ has limit $\frac37L$. But at the same time, $b_n=a_{n+1}$, so $L=\lim_n a_n=\lim_n b_n=\frac37L$ and $L=0$. Done.
Proving that the limit is zero if it exists? That's just limit arithmetic. Proving that the limit exists, on the other hand, requires more.
