Sorry Robin, I don't quite understand how you justify your first line:

This is clearly true if you apply

to a scalar, but it's not obviously true in general; in fact it's a statement of the result we're trying to show, for the particular choice of the coordinate connection (

).

I did it this way:

Noting that given any particular connection

, the action of

on an

arbitrary tensor

is

, we get:

But we have assumed our (given)

is torsion-free, so the third term in the brackets vanishes, and therefore:

for

all torsion-free connections

.