Towards a proof of Euler's identity

So here it goes: A sequence of complex numbers converges iff the sequences of real parts / imaginary parts converge. So let $(z_n)$ where $z_n =\left(1 +\frac{ir}{n}\right)^{n}$ where r is some real number. Now the sequence of real / imaginary parts will converge iff the sequence of absolute values converges. But since $z_n \bar{z_n} =\left(1 +\frac{ir}{n}\right)^{n}\left( 1 -\frac{ir}{n}\right)^{n} =\left(1 +\frac{r^2}{n^2}\right)^{n}$ , we have $\lim\limits_{n \to \infty} z_n\bar{z_n} =1$ . The sequence of real parts of $z_n$ is given by $\begin{aligned}{} \frac{\left(1 +\frac{ir}{n}\right)^{n} +\left(1 -\frac{ir}{n}\right)^{n}}{2} &=\sum_{k=0}^{\lfloor n/2\rfloor} r^{2k}(-1)^{k}\frac{\binom{n}{2k}}{n^{2k}} \end{aligned}$ So what I have in mind is to show that the above sum converges to $\cos{r}$ as n grows to infinity.

#Calculus

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