The discussion revolves around the uniform convergence of the sequence of functions defined as fn(x) = 0 if x ≤ n and fn(x) = x - n if x ≥ n, specifically on the intervals [a,b] and R. Participants are exploring the conditions under which uniform convergence occurs and the implications of the definitions involved.
Some participants have suggested that the sequence may be uniformly convergent on [a,b], while others express uncertainty about its behavior over the entire real line. There is an ongoing examination of definitions related to uniform convergence and the implications of the function's behavior as n increases.
Participants note that for sufficiently large n, the function fn(x) becomes zero over the interval [a,b], leading to discussions about the implications for uniform convergence. The distinction between behavior on [a,b] versus R is highlighted, with references to the Cauchy criterion for uniform convergence and the need to consider the entire real line.
Correct. Explicitly, if n > b, then f_n(x) = 0 for all x in [a,b].I thought that on the interval [a,b] f_n(x) and f(x) would equal 0 as n tends to infinity?
Let's start with "sup|f_n(x) - f(x)| exists". You can show that f_n -> 0 pointwise, correct? Pick your favorite number n. Does sup_x |f_n(x) - 0| exist?f_n converges to f uniformly if there exists an N, for all n>N, m_n := sup|f_n(x) - f(x)| exists and tends to 0 as n tends to infinity.
Yes, but only if your function is restricted to an interval [a,b] with b > 10. What happens when you let your function vary over the entire real line? Is there an upper bound on the difference |f_10(x)-0| = x-10 if you let x get arbitrarily large?So, say n=10, then sup | f_10(x) - 0 | = sup | x - 10 | = b-10 ?