So for a continuous section *f* , the real axis of Heath’s *V*-space is
the join of all the various sets *V*_{i, n} , and by the
Baire Category Theorem these cannot **all** be nowhere dense “in the reals”.
So at least one *V*_{i, n} is dense in some non-trivial real interval.

In such a *V*_{i, n} we can therefore find two *x*-axis points
that are closer together than 2/*n* , and for such points, their basic open
*V*s of finger-height 1/*n* ...

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