I have a doubt regarding the use of the Bolzano-Weierstrass theorem in a dynamic optimization problem. Exactly, the problem is the neoclassical growth model (or Ramsey model). In the website that I post below, in page 14, they invoke the aforementioned theorem so as to prove the existence of a solution. This is not the unic work that mention the theorem for the same purpouse. My question is, how can I use the B-W theorem so as to proof the existence of a solution in a such a casa?
The link to the website: https://homepage.ntu.edu.tw/~nankuang/Macro%20Theory%20III/Lecture%20Notes/The%20Neoclassical%20Growth%20Model%20in%20Discrete%20Time.pdf