in a triangle abc am is a median of abc and hence bm =mc. Proove that ab+bc+ca>2am

Answer 1

In any triangle, the sum of any two sides > the third side

so in triangle ABM:
AB + BM > AM
in triangle ACM
AC + MC > AM

adding:
AB + AC + BM+MC > 2AM
AB + AC + BC >2AM

Answer 2

btw, this is true not just for the median AM
In fact, the same steps can be used to prove it true if M is any point on BC.