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Extend the triangle ABC to a parallelogram such that AD=DK .
The triangles ADC and KDB are congruent (by SAS), so AC=BK .
Given that sum of the lengths of two sides of a triangles is greater than the third, we have
AK<AB+BK
AD+DK<AB+AC
2AD<AB+AC
Doing the same with the other medians,
we have,
2BE<BC+AB
2CF<AC+BC
Summing all three, we get,
2(AD+BE+CF)<2(AB+AC+BC)
or
AD+BE+CF<AB+AC+BC
Or in other words,
Sum of medians<Perimeter
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