If the current energy density of the universe is indeed dominated by a cosmological constant, then high-redshift sources will remain visible to us only until they reach some finite age in their rest frame. The radiation emitted beyond that age will never reach us due to the acceleration of the cosmic expansion rate, and so we will never know what these sources look like as they become older. As a source image freezes on a particular time frame along its evolution, its luminosity distance and redshift continue to increase exponentially with observation time. The higher the current redshift of a source is, the younger it will appear as it fades out of sight. For the popular set of cosmological parameters, I show that a source at a redshift $z_0\sim 5–10$ will only be visible up to an age of $\sim 4–6$ billion years. Arguments relating the properties of high-redshift sources to present-day counterparts will remain indirect even if we continue to monitor these sources for an infinite amount of time. These sources will not be visible to us when they reach the current age of the universe.