<s>[INST] Prove $\frac{1}{n} \sum^{n}_{k = 1} \left(1 + \frac{1}{k} \right) \geq (n + 1)^{\frac{1}{n}}$ Prove for every $n \in \mathbb{N}$ with $n \geq 1$ that
$$\frac{1}{n} \sum^{n}_{k = 1} \left(1 + \frac{1}{k} \right) \geq (n + 1)^{\frac{1}{n}}$$

What I've done so far:
Since $n \in \mathbb{N}_{\geq 1}$, it follows that $1 + \frac{1}{n} \in \mathbb{R}_{\geq 1}$. Then, by the AM-GM inequality we have
$$
\frac{1}{n} \sum^{n}_{k = 1} (1 + \frac{1}{k}) = \frac{1}{n}\left(1 + \frac{1}{1} + 1 + \frac{1}{2} + \cdot \cdot \cdot + 1 + \frac{1}{n} \right)
$$
$$
\geq \left[1 \cdot \frac{1}{1} \cdot 1 \cdot \frac{1}{2} \cdot \cdot \cdot 1 \cdot \frac{1}{n} \right]^{\frac{1}{n}}
$$
$$
= \left[\frac{1}{1} \cdot \frac{1}{2} \cdot \cdot \cdot \frac{1}{n} \right]^{\frac{1}{n}}.
$$
However, I'm not sure how to continue after this.
 [/INST] $$\frac{1}{n} \sum^{n}_{k = 1} (1 + \frac{1}{k})\geq  [(1 + \frac{1}{1})(1 + \frac{1}{2})  \cdot \cdot \cdot  (1 + \frac{1}{n})]^{\frac{1}{n}}$$ $$= [\frac{2 \cdot 3 \cdot \cdot \cdot (n+1)}{n!}]^{\frac{1}{n}}=(n+1)^{\frac{1}{n}}$$
</s>