Let $\mathcal{l}_\mathbb{R}^\infty$ be the space of bounded sequences in $\mathbb{R}$. We define a map $p: \mathcal{l}_\mathbb{R}^\infty\to\mathbb{R}$ by $$p(\underline x)=\limsup_{n\to\infty} \frac{1}{n}\sum_{k=1}^n x_k.$$ My notes claim that $$\liminf_{n\to\infty} x_n\le p(\underline x)\le \limsup_{n\to\infty} x_n.$$ I haven't found a neat way to show that this holds (only a rather complicated argument). Is there an easy, intuitive way ?
Average limit superior
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real-analysis
sequences-and-series
inequality
limsup-and-liminf
average