So for a continuous section f , the real axis of Heath’s V-space is
the join of all the various sets Vi, n , and by the
Baire Category Theorem these cannot all be nowhere dense “in the reals”.
So at least one Vi, n is dense in some non-trivial real interval.
In such a Vi, n we can therefore find two x-axis points
that are closer together than 2/n , and for such points, their basic open
Vs of finger-height 1/n ...
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