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}

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