I am struggling to understand what Question 1 and 2 are (In the Assignment) asking for and I cannot make a move to start off. For Question 2, I do not know how to get a j by itself after trying to minus 1/3 from aj. Thanks
2026-03-25 03:04:01.1774407841
Definition of Convergence of a Sequence and Series Questions
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Question 1: Let $(X_n)$ be a convergent sequence in $A_{X_0,N}$ with limit $X$.
Then we have $<N,X_n-X_0> \le 0$ for all $n$. The continuity of the inner product then gives $<N,X-X_0> \le 0$, hence $X \in A_{X_0,N}$.
Question 2 ,1):
Show that $|a_j-\frac{1}{3}|=\frac{8}{9j^2+3j} \le \frac{8}{9j^2} \le \frac{1}{j^2} \le \frac{1}{j} $.
Can you proceed ?