Prove sequence $a_n=n^{1/n}$ is convergent How to prove that the sequence $a_n=n^{1/n}$ is convergent using definition of convergence?

So here is an outline of a proof:
Step 1: Notice that $n^\frac{1}{n}\geq 1$ for all $n$.
Step 2: Prove that $a_n$ is monotonically decreasing for $n\geq 3$.  Equivalently we need to show that $n^{(n+1)}>(n+1)^n$.
Step 3: Show that there is a subsequence which converges to $1$.  I managed to do this by considering $b_n={a_{2^{2^n}}}$.  (It does not appear well in LaTeX as there are too many nested exponents.  I had typed this part out, but decided to remove it)
From these three facts you can conclude that the limit is $1$.
