MHB Showing that nth root of c_n is equal to nth root of c_n+1 in the limit

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Hello everyone!

I'm trying to show that $\lim \sup \sqrt[n]{c_{n+1}}=\lim \sup \sqrt[n]{c_n}$

This is my attempt:
$\lim \sup \sqrt[n]{c_{n+1}} = \lim \sup \sqrt[m-1]{c_m}=\lim \sup c_m \; ^{\frac{1}{m}}c_m \; ^{\frac{1}{m(m-1)}}$

I'm stuck here, I think I must use some exponential property that says that something decays faster than something or the ratio of two things is zero in the limit...

Any help is appreciated!
 
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OhMyMarkov said:
I'm trying to show that $\lim \sup \sqrt[n]{c_{n+1}}=\lim \sup \sqrt[n]{c_n}{n}$
What if $c_n=1$ for all $n$?
 
Ah excuse me LaTeX typo: I meant

I'm trying to show that: $\lim \sup \sqrt[n]{c_{n+1}} = \lim \sup \sqrt[n]{c_{n}}$

I fixed it in the thread
 
A sphere as topological manifold can be defined by gluing together the boundary of two disk. Basically one starts assigning each disk the subspace topology from ##\mathbb R^2## and then taking the quotient topology obtained by gluing their boundaries. Starting from the above definition of 2-sphere as topological manifold, shows that it is homeomorphic to the "embedded" sphere understood as subset of ##\mathbb R^3## in the subspace topology.
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