Let $\epsilon > 0$
$|1/n - L| < \epsilon$ for some $n\geq N$. (definition of convergence)
This implies $-\epsilon\cdot n < 1 - nL < \epsilon\cdot n$
Therefore $1 - nL > -\epsilon\cdot n$
Therefore $1 < n(\epsilon - L)$
Choose $\epsilon = 1/n$
Therefore $nL < 2$
Therefore $L < 2/n$ for all $n\geq N$
By Archimedean property, we know that this only holds when $L = 0$. Hence the sequence converges to $0$, by uniqueness of convergence the sequence does not converge to any $L > 0$.
My issue with this proof is that when I choose $\epsilon = 1/n$, technically $\epsilon$ is changing for every choice of $N$. Is this still a valid proof?