If $\triangledown$ is a connection, show that $\frac{1}{2}\triangledown$ does not satisify the product rule and is thus not a connection.

Question

If $\triangledown$ is a connection, show that $\frac{1}{2}\triangledown$ does not satisify the product rule and is thus not a connection.

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My attempt:

I'm a bit confused here on how to evaluate $\frac{1}{2} \triangledown_X fY$. Is the point of this to show that $\frac{1}{2} (\triangledown_X fY) \neq (\frac{1}{2} \triangledown)_X fY$?

$(\frac{1}{2} \triangledown)_X fY = XfY +f\frac{1}{2} \triangledown_X Y$

$\frac{1}{2} (\triangledown_X fY) = \frac{1}{2}(XfY +f \triangledown_X Y)$

And thus the product rule is not well defined with the connection $\frac{1}{2} \triangledown$.

I'd appreciate some insight on this one, thanks!