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?

Answer

Noticing that $n^\frac{1}{n} > 1$ for all $n$, it all comes down to showing that for any $\epsilon > 0$, there is a $n$ such that $(1+\epsilon) \geq n^\frac{1}{n}$, or by rearranging, that

$$(1+\epsilon)^n \geq n$$

Now, let's first of all choose an $m$ such that $(1+\epsilon)^{m}$ is some number bigger than 2, let's say the smallest number greater than $3$ that you can get. From here, swap $m$ for $2m$. This will make the left side a little over 3 times larger, and the right side 2 times larger. The next doubling will still double the right side, but the left side will increase roughly 9-fold. Repeating, we can easily see that the left side will at some point overtake the right side, and we have our $n$