Let be defined by

Let and

Show that with respect to , but is not convergent with respect to

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Now, I did the following, but it must be wrong since it doesn't depend on the choice of norm, so I get that in both cases.

Choose for some small n.

st but rather

st but rather

st

So, clearly, we have that st

So what have I missed that stops the above from working, and what is a better way of doing it?