Understanding the Properties of Tensor Products: A Demonstration

[Telemachus]

I have to demonstrate that if $$A^{rs}$$ is an antisymmetric tensor, and $$B_{rs}$$ is a symmetric tensor, then the product:
$$A^{rs}B_{rs}=0$$

So I called the product:
$$C^{rs}_{rs}=A^{rs}B_{rs}=-A^{sr}B_{sr}=-C^{rs}_{rs}$$
In the las stem I've changed the indexes, because it doesn't matters which is which, but I'm not sure this is fine (because I think r and s could have have associated differents values in the sum).

Then
$$2C^{rs}_{rs}=2A^{rs}B_{rs}=0$$
Is this ok?

 

[xaos]

interchanging and relabeling an implied double summation is okay.

 

[Telemachus]

Thank you xaos :)