Let $X$ be a compact and convex subset of $\mathbb R^n$ and let ${(x_{t})}_{t}$ be a sequence in $X$ such that $||x_{t+1}-x_{t}||_{\infty}\leq \frac{1}{t+1}$ for all $t=0,1,...$.
Let ${(x_{t_n})}_{t_n}$ be a convergent subsequence of $(x_t)_t$ such that $x_{t_n}\to \bar x$.
Is it the case that $\lim_{n\to \infty}\frac{1}{t_n+1}\sum_{\ell=0}^{t_n}x_{\ell} = \bar x$?
I am aware that Cesàro mean of a convergent subsequence converge to the limit of the subsequence when the average is taken only with respect to elements of the subsequence. Here I am averaging over all terms up to $t_n$ and taking the limit over indices in the subsequence.
EDIT:
The claim is false and a counterexample can be constructed using the sequence $\frac{0}{2},\frac{1}{2},\frac{2}{2},\frac{2}{3},\frac{1}{3},\frac{0}{3},\frac{1}{4},...$ and picking the subsequence $(x_{t_n})_{t_n}$ such that $x_{t_n}=0$. Then $\bar x=0$, but $\lim_{n\to \infty}\frac{1}{t_n+1}\sum_{\ell=0}^{t_n}x_{\ell}=\frac{1}{2}$. In fact, it seems like this is the Cesàro mean regardless of the subsequence.
So my new questions are:
Let $X$ be a compact and convex subset of $\mathbb R^n$ and let ${(x_{t})}_{t}$ be a sequence in $X$ such that $||x_{t+1}-x_{t}||_{\infty}\leq \frac{1}{t+1}$ for all $t=0,1,...$.
(1) **Is there always a convergent subsequence $(x_{t_n})$ with convergent mean $\lim_{n\to \infty}\frac{1}{t_n+1}\sum_{\ell=0}^{t_n}x_{\ell}=\bar x$? **
(2) **If $\lim_{n\to \infty}\frac{1}{t_n+1}\sum_{\ell=0}^{t_n}x_{\ell}=\bar x$, is it the case that $\lim_{t\to \infty}\frac{1}{t+1}\sum_{\ell=0}^{t}x_{\ell}=\bar x$? **