The limit of a sequence of points \ left ( x _ n : n \ in \ mathbb { N } \ right ) \; in a topological space " T " is a special case of the limit of a function : the domain is \ mathbb { N } in the space \ mathbb { N } \ cup \ lbrace + \ infty \ rbrace with the induced topology of the affinely extended real number system, the range is " T ", and the function argument " n " tends to + ", which in this space is a limit point of \ mathbb { N }.
12.
I'm happy with the equivalence of convergence under the given metric iff we get convergence for all the seminorms; so I guess my final question is about the correspondence between the limits and the topology : if we know ( as in this case ) that convergence in the induced topology from the seminorms is the same as convergence in a different topology ( such as the product topology here ), why does that necessarily mean the 2 topologies themselves are the same ? ( How do we know there isn't some weird collection of open sets, not the same as the product topology, which gives us convergence in the seminorms iff we have convergence in the product topology?
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