To prove sample mean is an unbiased estimator of the population mean, we use the following step, $\mathbb{E}[\sum \frac{X_i}{n}] = \frac{1}{n} \sum \mathbb{E}[X_i] = \mu$, given that $\mathbb{E}[X_i] = \mu$.
Now my question is if we take say first random variable $X_1$ as an estimate we can go ahead and say that since $\mathbb{E}{[X_1]} = \mu$, $X_1$ is an unbiased estimator. Does that mean $X_1$ or for that matter any value is an unbiased estimator?
I know it is not. I am doing a conceptual mistake in thinking, please let me know where I am going completely wrong.
[Edit] As commented below for iid's every sample is indeed an unbiased estimate of the mean.