Does it make sense to describe the SEM as an estimate? If one calculates a standard error of the mean from the sample mean and sample standard deviation of n observations drawn from a normally distributed population, does it make sense to call that SEM an estimate of some "true" SEM?
Thus, if I increase n, is it true that I not only reduce the size of my SEM, but also increase the accuracy of my estimate (because the sample standard deviation is better estimate)?

Consider: $s_x$ (or $\widehat{\sigma}_{x}$, if you prefer) estimates $\sigma$ (with bias), and in the same fashion $s_{\overline{x}}$ (or $\widehat{\sigma}_{\overline{x}}$) estimates $\sigma_{\overline{x}}$. Right? $\frac{s_{\overline{x}}}{\sqrt{n}}$ estimates $\frac{\sigma_{\overline{x}}}{\sqrt{n}}$?
