Proof of extended version of non-random "almost supermartingale" In this question, a non-random version of "almost supermartingale" theorem is proved.
Here, I would like to extend/apply the non-random version to the slightly different situation. I wonder whether the theorem holds under this extended version case, does it?
Let me define some non-negative variables
\begin{align}
v_k &:= \alpha_k\|x_k - x^\star \|_2^2 \\
t_k &:= \gamma_k \|x_k - x_{k-1} \|_2^2
\end{align}
where $\alpha_k, \gamma_k \in \mathbb{R}_{+}$.
ADDENDUM1: $\{x_k \!\in \mathbb{R}^d\!\}$ are the sequences generated by a convex optimization algorithm and $x^{\star} \!\in \mathbb{R}^d$ is an optimal solution.
ADDENDUM2: {Both $\alpha_k$ and $\gamma_k$ are monotonically decreasing over increasing $k$, i.e., $\alpha_{k+1} \leq \alpha_{k}$ and $\gamma_{k+1} \leq \gamma_{k}$ forall $k$. The supremum of both $\{\alpha_k\}$ and  $\{ \gamma_k\}$ is less than infinity.}}
Let $\beta_k \geq 0$, which satisfy $\sum_{k=0}^\infty \beta_k < \infty$. Also, let $\{s_k\}$ be another non-negative variable.
Assume
\begin{align}
v_{k+1} + t_{k+1} \leq \left( 1 + \beta_k \right) v_{k} + t_{k} - s_k \tag{$\clubsuit$},
\end{align}
Question:
Then, can we extend or/and apply non-random version of "almost supermartingale" theorem (or some other theorem?) to $(\clubsuit)$ such that
\begin{align}
v_k &\rightarrow v^{\infty} \ \text{or} \ x_k \rightarrow x^{\star} ?
\end{align}
and
\begin{align}
t_k &\rightarrow 0 ?
\end{align}
 A: $\newcommand\R{\mathbb R}$The answer to each of your three questions is no.
Indeed, suppose that $\alpha_k=\gamma_k=1$ and $\beta_k=s_k=0$ for all $k$. Suppose that the dimension $d$ is $1$ (then the counterexample presented below can be obviously "imbedded" into $\R^d$ for any natural $d$).
Let $x^\star:=0$. Let
\begin{equation}
    (x_0, x_1, x_2, x_3):=(2-\sqrt{5},-1,\sqrt{5}-2,1)
\end{equation}
and then let
\begin{equation}
    x_k:=x_{k-4}
\end{equation}
for natural $k\ge4$, so that the sequence $(x_k)_{k\ge0}$ is periodic with period $4$.
Hence, the sequences $(v_k)_{k\ge0}=(x_k^2)_{k\ge0}$ and $(t_k)_{k\ge1}=((x_k-x_{k-1})^2)_{k\ge1}$ are also periodic with period $4$, with
\begin{equation}
    (v_1, v_2, v_3,v_4)=(v_1, v_0, v_1,v_0), 
\end{equation}
\begin{equation}
    (t_1, t_2, t_3,t_4)=(t_1,t_0, t_1,t_0),
\end{equation}
\begin{equation}
    v_0=9 - 4\sqrt5,\quad v_1=1,\quad t_0:=6-2\sqrt5,\quad t_1=14 - 6\sqrt5, 
\end{equation}
so that $   v_1+t_1=v_0+t_0$ and hence
\begin{equation}
    v_{k+1} + t_{k+1}=v_k+t_k \tag{$\clubsuit\clubsuit$}
\end{equation}
for all natural $k$. So, recalling that $\beta_k=s_k=0$ for all $k$, we see that your condition ($\clubsuit$) holds.
However, since each of the sequences $(v_k)_{k\ge0}$, $(x_k)_{k\ge0}$, and $(t_k)_{k\ge1}$ is non-constant and periodic, we see that none of the conditions $v_k\to v^{\infty}$, $x_k \to x^{\star}$, $t_k\to0$ holds.
Remark: If we want the sequence $(v_k)_{k\ge0}$ to be alternating between two different values and if we want $(\clubsuit\clubsuit)$ to hold for all natural $k$, then
the sequence $(x_k)_{k\ge0}$ in the above example is uniquely determined up to rescaling (that is, replacing the sequence $(x_k)_{k\ge0}$ by the rescaled sequence $(cx_k)_{k\ge0}$ for some nonzero real $c$) and/or the swapping $x_{2j}$ with $x_{2j+1}$ for all $j=0,1,\dots$, to get the sequence $(x_1,x_0,x_3,x_2,\dots)$.

The picture below shows the oscillatory behavior of the periodic sequence
$$
\begin{aligned}
(x_k)_{k\ge0}&=(x_0, x_1, x_2, x_3, x_0, x_1, x_2, x_3,\dots) \\ 
&=(x_0, x_1, -x_0, -x_1, x_0, x_1, -x_0, -x_1,\dots)\,:
\end{aligned}
$$

