How does one go about proving that the limit of i^p as p approaches infinity is equal to the force of interest?

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To prove that the limit as p appraoches infinity of i^(p) you can use Eulers rule.

Recall that the $$\lim\limits_{n \rightarrow \infty} (1 + \frac{x}{n})^n = e^x $$

By definition you also know that $$(1+i) = (1+\frac{i^(p)}{p})^p$$

therefore $$\lim\limits_{n \rightarrow \infty} (1 + \frac{i^(p)}{p})^p = e^{i(\infty)}$$

and so $$ i^{\infty} = \delta$$