A confusing sequence question

Hello, I am completely lost with this question:

Let $\displaystyle x_n = n^{\frac{1}{n}}$ for $\displaystyle n \in \mathbb{N}$.

1. Show that $\displaystyle x_{n+1} < x_n$ if and only if $\displaystyle (1+ \frac{1}{n}) < n$ and infer that the inequality is valid for $\displaystyle n \geq 3$. Conclude that $\displaystyle (x_n)$ is ultimately decreasing and that $\displaystyle x = \lim(x_n) $exists.

2. Use the fact that the subsequence $\displaystyle (x_{2n})$ also converges to $\displaystyle x$ to conclude that$\displaystyle x=1$.

I see why $\displaystyle (1+ \frac{1}{n}) < n$ given that $\displaystyle n \geq 3$ , since $\displaystyle \lim (1+ \frac{1}{n}) = e < 3$

But that's pretty much what I got =(