Why is the restricted direct product topology on the idele group stronger than the topology induced by the adele group?

I am confused by the saying that the restricted direct product topology on the idele group is stronger than the topology induced by the adele group. And perhaps this is elaborated by the following example

Let $p_n$ be the $n$-th positive prime in $\mathbb{Z}$, and let $\alpha^{n}=(\alpha^{(n)}_v)\in\mathbb{A}_\mathbb{Q}$ with $\alpha^{(n)}_v=p_n$ if $v=p_n$ and $\alpha^{(n)}_v=1$ if $v\neq p_n$. The result is a sequence $\{\alpha^{n}\}$ of ideles in $\mathbb{I}_\mathbb{Q}$. Then this sequence converges to the idele $(1)_v$ in the topology of the adeles but not converges in the topology of the ideles.

Any comment is appreciated!

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2022-07-25 20:44:17
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