I am learning about the idea of estimators in statistics and understand that the role of an estimator is to estimate a parameter of the population distribution. In my book it says that the estimation function for the population mean is the sample mean, and there is a proof of why this is, but I don't understand this.
Here is a link to something showing the same proof. This starts with the expectation of the sample mean and substituting our formula for the sample mean into there and then moving 1/n out of the summation and then saying this is the same as the sum over the expectation values of the individual data points. They then say this is the same as n time the population mean.
I thought the point of this was to show that the expectation of the sample mean is the population mean, but it looks like we are just asserting this rather than showing it. We are saying the expectation of our data point is the population mean.
Can someone explain this to me?