It's not a metric topology, basically because there are uncountably many of the defining seminorms
. A basic family of neighbourhoods of f would consist of sets of the form
for each finite set of points
It is called the topology of pointwise convergence because a sequence (or more generally a directed net)
converges to the limit
iff
for each point
For the last part of the question, I think that you have to use a cardinality argument. You want to find a sequence
in
which converges to 0 (in the topology of pointwise convergence) so slowly that if you multiply its terms by any sequence of scalars that tends to infinity then the resulting sequence
no longer converges to 0.
The set
of all sequences of
integers that tend to infinity has cardinality
, which is the same as the cardinality of [0,1]. So there exists a bijective map
. Define the function
by
.
Then
for all
, but
Finally, I leave you to check that if
fails to converge to 0 for all sequences of integers tending to infinity, then the same will be true for all sequences of reals tending to infinity.