Is the following result true :
Let $(a_n)_{n\in\mathbb{N}}$ is a sequence of nonnegative integers that doesn't diverge, then the sequence is $T$-periodic, with $T\ge 1$ an integer.
2026-03-28 02:04:47.1774663487
Non divergence of a sequence of integers
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It depends on what you mean by non-divergence. If you mean that it does not diverge to infinity then this is a simple counterexample: $$1, 0, 1, 0, 0, 1, 0, 0, 0, 1, \ldots$$
If you mean that it converges (in the usual metric) then it will eventually be constant. Periodic after that point but not necessarily periodic from the beginning. Example: any finite random sequence followed by a single number repeating for ever.