The discussion centers on the properties of a nonnegative nonincreasing sequence $\{p_n\}$ that converges to a limit $p \ge 0$, and the behavior of a nondecreasing function $f : [0,\infty) \to [0,\infty)$. It is established that since $\{p_n\}$ is nonincreasing, $p = \inf\{p_n : n \in \mathbb{N}\}$, leading to the conclusion that $f(p_n) \ge f(p) \ge 0$. The participants clarify that the inequalities are not strict, and a misunderstanding regarding the relationship $p_{n+1} \le p_n - f(p)$ is addressed, confirming it is not necessarily true.
PREREQUISITESMathematicians, students of real analysis, and anyone interested in the behavior of sequences and functions in mathematical contexts.
Euge said:Hi ozkan12,
Since $\{p_n\}$ is nonincreasing and converging to $p$, $p = \inf\{p_n : n\in \Bbb N\}$, so $p_n \ge p$ for all $n\in \Bbb N$. Consequently, as $f$ is nondecreasing, $f(p_n) \ge f(p)$. So we have $f(p_n) \ge f(p) \ge 0$.
Note that the inequalities above are not strict.
ozkan12 said:ok thanks a lot :) but I will ask one question : so, pn+1<=pn-f(p)...how this happened ?
Euge said:I'm sorry, I don't understand -- where did you get $p_{n+1} \le p_n - f(p)$? This need not be true.