The discussion revolves around the concept of monotonic sequences in the context of providing examples of two monotonic sequences whose sum is not monotonic. Participants explore definitions and properties of monotonicity, questioning the nature of sequences and their sums.
The discussion is active with multiple interpretations of monotonicity being explored. Some participants provide examples and counterexamples, while others engage in clarifying the definitions and properties of monotonic sequences. There is no explicit consensus on the definitions being debated.
Participants reference various mathematical sources and definitions, indicating a lack of agreement on the terminology and properties of monotonic sequences. The discussion reflects differing interpretations based on various textbooks and resources.
If the terms of the sequence are a subset of an ordered set, then a monotonically increasing sequence is one for which each term is greater than or equal to the term before it; if each term is strictly greater than the one preceding it, the sequence is called strictly monotonically increasing. A monotonically decreasing sequence is defined similarly. Any sequence fulfilling the monotonicity property is called monotonic or monotone. This is a special case of the more general notion of monotonic function.
In calculus, a function f defined on a subset of the real numbers with real values is called monotonic (also monotonically increasing, increasing, or non-decreasing), if for all x and y such that x ≤ y one has f(x) ≤ f(y), so f preserves the order. Likewise, a function is called monotonically decreasing (also decreasing, or non-increasing) if, whenever x ≤ y, then f(x) ≥ f(y), so it reverses the order.
Yes, and irrelevant. That talks about what is true for strictly increasing or decreasing sequences which was not in question here. The question was about monotonic sequences and there is nothing that requires they be strictly increasing or decreasing.nuclearrape66 said:no...wikipedia says this..."Functions that are strictly increasing or decreasing are one-to-one (because for x not equal to y, either x < y or x > y and so, by monotonicity, either f(x) < f(y) or f(x) > f(y), thus f(x) is not equal to f(y))."