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This is what I have to prove:

$\lim_{n\rightarrow\infty}\inf a_n\leq\lim_{n\rightarrow\infty}\inf(\frac{1}{n}\sum_{i=1}^na_i)\leq\lim_{n\rightarrow\infty}\sup(\frac{1}{n}\sum_{i=1}^na_i)\leq\lim_{n\rightarrow\infty}\sup a_n$

Note, that $a_n$ and $b_n$ are bounded sequences. I'm gonna be totally honest: I have no clue where to even start with this proof. Can someone give me some advice?

LinearAlgebruh
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  • You should start by understanding the definition of liminf and limsup. For example suppose $\liminf a_n=A$ then what does it mean for values of sequence $a_n$ for large $n$? – Paramanand Singh Nov 10 '19 at 09:32
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    Note also the symmetry: The “$\liminf$ inequality” follows from the “$\limsup$ inequality” by considering $(-a_n)$ instead of $(a_n)$. – Martin R Nov 10 '19 at 09:48

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