I've been trying to figure this out for a while and I'm totally lost.

My goal is to show that for two density operators $p$, $q$, that $$||p^{\otimes n} - q^{\otimes n}|| \leq n ||p-q||$$

So far I have constructed a weak inductive hypothesis over n to get

$$||p^{\otimes (n-1)} - q^{\otimes (n-1)}|| \leq (n-1) ||p-q||$$ However, I'm having a great deal of difficulty manipulating my tensor products on the LHS to be able to use my IH. What first steps can be taken?

Note: the norm being used here is the trace norm

I've been trying to figure this out for a while and I'm totally lost.

My goal is to show that for two density operators $p$, $q$, that $$||p^{\otimes n} - q^{\otimes n}|| \leq n ||p-q||$$

So far I have constructed a weak inductive hypothesis over n to get

$$||p^{\otimes (n-1)} - q^{\otimes (n-1)}|| \leq (n-1) ||p-q||$$ However, I'm having a great deal of difficulty manipulating my tensor products on the LHS to be able to use my IH. What first steps can be taken?

Note: The norm being used here is the trace norm.