Let,
n1 = MST edges = V-1 = n-1 and
n2 = Rest of the edges
(MSTedgesSum + RestedgesSum ) = (n1 + n2) Aavg
MSTedgesSum = (n1 + n2)Aavg - RestedgesSum
MSTedgesSum = (n1)Aavg + n2(Aavg - AavgRestEdges)
Thus, Mst edges Sum can atmost be equal to (n1)Aavg if ,
n2(Aavg - AavgRestEdges ) =0
Aavg - AavgRestEdges can take zero, positive as well as negative value and
n2 =0 is possible only when we have graph G itself being MST and there are no extra edges
Thus, MSTedgesSum >= (n1)Aavg. , when Aavg - AavgRestEdges takes positive value.
MSTedgesSum <= (n1)Aavg , when Aavg - AavgRestEdges takes negative value.
Equality will hold when graph G = MST.
But the question demands MSTedgesSum <= (n1)Aavg. is always true. So, the stmt. false.