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--- pow:problem4s25 [2025/04/02 18:44] – mazur
+++ pow:problem4s25 [2025/04/03 16:51] (current) – mazur
@@ Line 1: / Line 1: @@
+<box 85% round orange| Problem 4 (due Monday, March 31 )>
+A sequence $a_1,a_2,\ldots$ of real numbers has the following properties:
+(i) $|a_1+a_2+\ldots +a_k|\leq 1$ for every $k$;
+(ii) $|a_k-a_{k-1}|\leq 1/k$ for every $k\geq 2$.
+Suppose that $\displaystyle |a_k|\geq \frac{c}{\sqrt{k}}$ for infinitely many $k$. Prove that $c\leq \sqrt{2}$.
+</box>
+We received a solution from Josiah Moltz and Dr Mathew Wolak. For detailed solutions see the following link {{:pow:2025sproblem4.pdf|Solution}}.