The question is very simple. Is it true that $\text{limsup}|a_n|^{1/n}\geq \text{limsup}|a_{2n}|^{1/n} $ where $<a_n>$ is some real sequence? Can this be proved or disproved? And also what if I make the following change to the inequality; $\text{limsup}|a_n|^{1/n}\geq \text{limsup}|a_{2n}|^{1/2n} $. Does it make a difference to the answer? Hope someone can help me out. Thanks
2026-03-27 16:27:43.1774628863
$\text{limsup}|a_n|^{1/n}\geq \text{limsup}|a_{2n}|^{1/n} $
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$$\limsup|a_n | ^{1/n} ≥\limsup|a_{ 2n} | ^{1/2n} $$
is true. The other one is not.
This is because any subsequence of $( u_{2n})$ is also a subsequence of $(u_n)$.