The easiest way is to implement a Turing machine in lambda calculus. This is quite simple, because Lambda Calculus is practically a high-level programming language. The advantage of this approach is that it does not require any other mathematical dependencies, and therefore it should provide the easiest way to present your argument.
In terms of mathematical proof, the shortest way is to implement another paradigm, which, as has already been shown, is a complete Turing, like μ-recursive functions. They are already recursively defined, so their expression in lambda calculus is a bit more elegant than the Turing machine itself.
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