I have been trying to understand exactly how would would formally demonstrate the equivalence between uniform stability and Lyapunov stability for fixed points of autonomous systems.
First, apologies if I commit some formatting error or other faux-pas here on my first post.
I have found the excellent explanation given here, https://math.stackexchange.com/q/2353040 , by M3053. I find this explanation intuitively satisfying, but I have been struggling without success to rigorously demonstrate this equivalence for myself. Moreover, all the references I have consulted so far seem content to state this equivalence as an obvious fact which unfortunately has not been so trivial for me to establish.
If anyone would be willing to elaborate upon the above explanation further, and provide some insight as to how it would be appropriate to go about demonstrating this equivalence, I would be grateful.