Prove that the sequence $a_{1}= 1$, $a_{n+1} = \sqrt[n]{a_{1}+\dots+a_{n}}$ is bounded below

So I have a sequence $a_{n+1} = \sqrt[n]{a_{1}+\dots+a_{n}}$, $n \in \mathbb{N}$ and where is $a_{1}= 1$.

I have to prove that there is $c > 0$ such that for every $n \in \mathbb{N}$, $a_{n} \geq c$ holds.

I began proving this with mathematical induction and I came to this point for $n \rightarrow n+1$$$a_{1}\dots+a_{n} \geq c^n$$ and I do not know how to finish it.

Also I have to prove that this is increasing sequence and I came to this point $$a_{1}+\dots+a_{n}+a_{n+1}\geq a_{1}+\dots+a_{n}$$ and from this I get that $$a_{n+1}\geq0$$

If is this even correct so far?

Any help?