Thin copper wire does not keep stretching for ever. In general it only
stretches once under a given tension.

Copper has the very useful property of work-hardening as it stretches. If
the tension is not increased further than its previous maximum tension it
remains at the same length.

If any part of a length has a tendency to stretch more than another, perhaps
because it is slightly thinner, then that part hardens first until the
remainder catches up.

Give the wire a good stretch, to make it dead sraight, before erecting an
antenna and it will remain at that length. Electrical properties will not be
significantly affected.

I have used the same thin 18 and 20 gauge, enamelled, magnet, antenna wire
for many years without any problems of this sort.
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Reg, G4FGQ