I'm having some problems proving that the variance of an average of $N$ iid random variables is equal to $\frac{1}{N}\text{Var}[X_1]$, where $X_1$ is one of the considered random variables. Formally, this is the statement I want to prove:
Let $(X_i)_{1 \leq i \leq N}$ be a sequence of iid random variables. Define $m^N=\frac{1}{N}\sum_{i=1}^N X_i$. Then we have: $$\text{Var} [m^N]=\frac{1}{N}\text{Var}[X_1].$$
I'm familiar with the proof of the form-->
$$\text{Var} [m^N]=\text{Var}[\frac{1}{N}\sum_{i=1}^N X_i]=\frac{1}{N^2}N \cdot \text{Var}[X_1]=\frac{1}{N}\text{Var}[X_1],$$
in which the basic variance properties are used, but I'm trying to prove the statement in a bit different way and I just can not see what I'm doing wrong. This is what I've got going-->
\begin{equation} \begin{split} \text{Var}[m^N]&=\text{Var}[m^N-\mathbb{E}[X_1]]\\ &=\mathbb{E}[(m^N-\mathbb{E}[X_1])^2]-\left(\mathbb{E}[m^N-\mathbb{E}[X_1]]\right)^2 \\ & = \mathbb{E} \left( (m^N)^2-2m^N \mathbb{E}[X_1]+(\mathbb{E}[X_1])^2 \right)-\left( \mathbb{E}[m^N] - \mathbb{E}[X_1] \right)^2 \\ & = \mathbb{E}[(m^N)^2]-2\mathbb{E}[m^N]\mathbb{E}[X_1]+(\mathbb{E}[X_1])^2-\left( \mathbb{E}[m^N] - \mathbb{E}[X_1] \right)^2 \\ & \text{Since }\mathbb{E}[m^N]=\mathbb{E}[X_1] \text{ we next have-->} \\ & = \mathbb{E}[(m^N)^2]-\left(\mathbb{E}[X_1]\right) ^2\\ & = \mathbb{E}[\left( \frac{1}{N}\sum_{i=1}^N X_i \right) ^2]-\left(\mathbb{E}[X_1]\right) ^2 \\ & = \frac{1}{N^2}\mathbb{E}[ \left( \sum_{i=1}^N X_i \right) ^2]-\left(\mathbb{E}[X_1]\right) ^2 \\ & = \frac{1}{N^2}\mathbb{E}[ \sum_{i=1}^N \left( \ X_i \right) ^2 +\sum_{i,j=1; i\neq j}^N X_iX_j]-\left(\mathbb{E}[X_1]\right) ^2 \\ & = \frac{1}{N^2}\left[ \mathbb{E}\left(\sum_{i=1}^N \left( \ X_i \right) ^2 \right)+ \mathbb{E} \left( \sum_{i,j=1; i\neq j}^N X_iX_j \right) \right]-\left(\mathbb{E}[X_1]\right) ^2 \\ & = \frac{1}{N^2}\left[ \sum_{i=1}^N \mathbb{E} \left[ \left( \ X_i \right) ^2 \right] + \sum_{i,j=1; i\neq j}^N \mathbb{E} [ X_iX_j] \right]-\left(\mathbb{E}[X_1]\right) ^2 \\ & = \frac{1}{N^2}\left[ N \cdot \mathbb{E} \left[ \left( \ X_1 \right) ^2 \right] + (N^2-N) \cdot \mathbb{E} [ \left ( X_1 \right) ^2] \right]-\left(\mathbb{E}[X_1]\right) ^2 \\ & = \frac{1}{N^2}\left[ N^2 \cdot \mathbb{E} [ \left ( X_1 \right) ^2] \right]-\left(\mathbb{E}[X_1]\right) ^2 \\ & = \mathbb{E} [ \left ( X_1 \right) ^2] -\left(\mathbb{E}[X_1]\right) ^2 \\ & = \text{Var} [X_1] \text{ ???} \end{split} \end{equation}
Thanks!