It is known that a turing machine and the lambda calculus are equivalent in power. I now want to try to prove this myself. I think proving that the lambda calculus is at least as powerful as a turing machine should be relatively easy, by trying to simulate a turing machine in lambda calculus. This should be possible by storing the tape as a list with some encoding, and encoding a transition function as a lambda term.
However, I'm having trouble with proving that a turing machine is at least as powerful as lambda calculus. Any pointers on how to start trying?