I have the sequence a non-negative sequence $u_k$ that satisfy the following relation $$ u_{k+1} \leq u_k + \alpha \sum_{i=1}^d (\Delta_{i,k+1})^2$$ where $\alpha > 0$, $d \in \mathbb{N}$ (fixed) and $\Delta_{i,k+1}$ is a non-increasing non-negative sequence with $\underset{k \rightarrow \infty} {\lim} \Delta_{i,k+1}= 0, \forall i \in \mathbb{N}$.
I'm looking to find an upper bound for $u_k$. From the above, I can write $$ u_{k+1} \leq u_0 + \alpha \sum_{i=1}^d \left((\Delta_{i,k+1})^2 + \dots + (\Delta_{i,1})^2 \right)$$ If I impose that $ \sum_{k=0}^{\infty} (\Delta_{i,k+1})^2 < \infty, \forall 1 \leq i \leq d$, then I could say that $u_k$ can be upper bounded by a certain constant $C$. Is there a way to prove things without this assumption or at least relax it?
Thanks.