Bound of a sublinear function defined using lim sup

Question

In my functional analysis course, we defined the sublinear function on $l^{\infty}_{\mathbb{R}}$: $$p(x)=\limsup_{n\rightarrow\infty}\frac{1}{n}\sum_{k=1}^n x_k .$$ What I don't get is why we have: $\forall m\in\mathbb{N}$, $$p(x)=\limsup_{n\rightarrow\infty}\frac{1}{n-m+1}\sum_{k=m}^n x_k$$ and why that implies $p(x)\in[\inf\limits_{k\geq m}x_k, \sup\limits_{k\geq m}x_k]$

Answer 1

The question has already been answered by mathworker21.

In particular, the first assertion follows from the equality $$ \frac{1}{n}\sum_{k=1}^n x_k = \frac{1}{n}\sum_{k=1}^{m-1} x_k + \frac{n-m+1}{n} \cdot \frac{1}{n-m+1}\sum_{k=m}^n x_k. $$

At r.h.s., as $n\to +\infty$ (and keeping $m$ fixed) the first summation goes to $0$, whereas $\frac{n-m+1}{n} \to 1$.