Let $(M_n)_{n=0}^{\infty}$ be a strictly increasing sequence of real numbers such that $M_0=0$, and the sequence is bounded above by $M>0$. Define the minimization problem $$\min_{\zeta} \sum_{n=0}^{\infty}M_nc^n$$ where $c$ is a fixed real number such that $0<c<1$, and $\zeta$ is the set of all sequences $(M_n)_{n=0}^{n=\infty}$ that satisify the above conditions. Find the minimum and the minimizor, and if they do not exist, find the infimum of this expression.
I came up with the minimization problem by myself and I was trying to solve it, but I cannot gurantee that the solution exists. So I started to consider the infimum, which should exist since the expression is bounded below by $0$, but I do not know how find out what is the infimum and if it is attainable. My knowledge about math is limited to Baby Rudin, so if more advanced knowledge might be helpful, it would be helpful if you can provide a link about it. Any insights would be helpful. Thanks in advance.
If $(M_n)$ is strictly increasing bounded from above by $M$ and $0<c<1$ we get that $$\sum_{n=0}^{\infty}M_nc^n<M \sum_{n=0}^{\infty}c^n=\dfrac{M}{1-c}.$$
For any $\epsilon \in (0,1)$ we have that $(\epsilon M_n)$ is strictly increasing
$$\sum_{n=0}^{\infty}\epsilon M_nc^n<\dfrac{\epsilon M}{1-c}.$$
This shows that
$$\inf_{\zeta} \sum_{n=0}^{\infty}M_nc^n=0.$$