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The main point in any security proof for the one-time pad is that xoring a plaintext with a (uniformly) random bit string yields a (uniformly) random bit string no matter what the plaintext was. Therefore, picking some $C$ uniformly at random (as your lecturer suggested) is, in highly informal terms, essentially equivalent to being given that very same $C$ ...
The proof for the perfect secrecy property of the one time pad is quite simple. It makes use of basic probabilities and it says that: $$Pr[M=m|C=c]=Pr[M=m]$$ for a probability distribution M$\{0,1\}^n$ for the message space and a probability space C for the ciphertext space. Proof: Pr[C=c]=\sum{Pr[C=c|M=m']\cdot Pr[M=m']} =\sum{Pr[K=m'\oplus c]}\cdot ...