It is clear to me because of the definition of lim sup, but is there a formal way to show that?
How does one justify $\limsup\limits_{n\rightarrow \infty} a_n=\limsup\limits_{n\rightarrow \infty} a_{n+1}$?
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sequences-and-series
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0What is your definition of limsup? – 2017-01-09
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0prove it by contradiction. suppose they are different and find out that you have proposed an absurd – 2017-01-09
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0@PedroTamaroff limsup of $x_n$ is the upper bound of $x_n$ – 2017-01-09
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1$\lim \sup a_n = \lim_{n \to \infty} b_n$, where $b_n = \sup \{a_k: k \ge n\}$. So you just need to prove that if a sequence $(x_n)$ is convergent or $\to \infty$, then $\lim x_n = \lim x_{n+1}$. – 2017-01-09
1 Answers
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Let $A_n = \sup_{k \ge n} a_k$. Then by definition:
$$\limsup_{n \to \infty} a_n=\lim_{n \to \infty}A_n$$
Moreover:
$$\limsup_{n \to \infty} a_{n+1}=\lim_{n \to \infty}A_{n+1}$$
So you just need to prove $\lim_{n \to \infty}A_n=\lim_{n \to \infty}A_{n+1}$. This can be done using your favorite definition of a limit.