Proof check that lim sup $(c_n^{\frac{1}{n}})_{n=m}^{\infty} \leq$ lim sup $(\frac{c_{n+1}}{c_n})_{n=m}^{\infty}$

Let $(c_n)_{n=m}^{\infty}$ be a sequence of positive numbers.

$L$ := lim sup $(\frac{c_{n+1}}{c_n})_{n=m}^{\infty}$. Choose any $\epsilon > 0$, this implies that for some $N \in \mathbb{N}$ we have for all $n \geq N, \frac{c_{n+1}}{c_n} \leq L + \epsilon$. We can show by a trivial induction that this implies $c_{N+k} \leq (L + \epsilon)^{k}c_N$ for all $k \in \mathbb{N}$, and so $c_{N+k}^{\frac{1}{N+k}} \leq (L + \epsilon)^{\frac{k}{N+k}}c_N^{\frac{1}{N+k}}$ for all $k \in \mathbb{N}$

Case 1: $L \geq 1$. Then we can find a $k \geq N$ such that $c_N^{\frac{1}{N+k}} \leq 1 + \epsilon \leq (L + \epsilon)$ so $c_{N+k}^{\frac{1}{N+k}} \leq L + \epsilon$.

Hence, we can find a $n \in \mathbb{N}$ such that $c_n^{\frac{1}{n}} \leq L + \epsilon$. We now set $N = n$.The claim now is $(L + \epsilon)^{\frac{k}{N + k}}c_N^{\frac{1}{N+k}} \leq L + \epsilon$ for all $k \in \mathbb{N}$. The base case is easy, since $c_N^{\frac{1}{N}} \leq L + \epsilon$.

Assume that $(L + \epsilon)^{\frac{k}{N + k}}c_N^{\frac{1}{N+k}} \leq L + \epsilon$ for an arbitrary $k$. Then $(L + \epsilon)^{k}c_N \leq (L + \epsilon)^{N+k}$ Multiply both sides by $L + \epsilon$ so $(L + \epsilon)^{k+1}c_N \leq (L + \epsilon)^{N+k+1}$. Take the $N + k + 1$th root of both sides to complete the proof.

Otherwise, if $L < 1$, we have that $c_n$ is strictly decreasing beyond $N$, and therefore the sequence converges (bounded below by zero) and so is bounded. This means $c_n^{\frac{1}{n}}$ converges to something between 0 and 1.

Since $c_{N+k}^{\frac{1}{N+k}} \leq (L + \epsilon)^{\frac{k}{N+k}}c_N^{\frac{1}{N+k}}$ we take limit supremum of both sides as k tends to infinity and see that lim sup $c_n^{\frac{1}{n}} \leq L + \epsilon$ for all $\epsilon > 0$, and so this case is finished.

Therefore we can conclude that $L + \epsilon$ is eventually an upper bound on the sequence $c_n^{\frac{1}{n}}$ for all $\epsilon > 0$, and therefore the limit superior of that sequence is less than or equal to L.

Please let me know if I have made any mistakes, and what are some optimal ways to approach this question. Thank you in advance.