I recently came across the notion of an accumulation point as a result of a certain gradient descent variation. The following definition was found:
An accumulation point $P$ is such that there are an infinite number of terms of the sequence in any neighborhood of $P$.
I'm not sure if I understand the above definition correctly. Since the gradient of the function at this point needs not to be zero ($0$), but is zero at the stationary point, could one state that a stationary point is an accumulation point if it is a result of an iterative approach of minimization of a function $f(\cdot)$, satisfying $$f(P)-f(P_1)=0,$$where $P_1$ is the next iterate?
Note that the notion of oscillation adds to the confusion. Namely, it may not hold that $P=P_1$, and the minimization approach switches between the two (if the termination is determined by relative change of $P$ ). How could the accumulation point be defined in this case? (the last is inspired by the cluster point definition I came across; so, a connection between the three would be appreciated: accumulation point, oscillation, cluster point)
Since it is stated here (Theorem 8.5) that a convergent sequence has one and only one limit point (accumulation point), what could be induced from the case with oscillations? In other words, could one state that the sequence is actually non-convergent?