Say you dropped a mirror into a black hole while observing at a distance and holding a clock such that the clock's face was pointing to the black hole. What is the latest time you would view on the reflection of the clock? What time would the reflection of the clock show after waiting for a long period of time.
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The first and second inequality/equation in ProfRob's answer here is the answer to your question. I'm posting this as an answer instead of a suggested duplicate because the questions are distinct and parts of the answers are unrelated to your question.
The image reflected in the mirror asymptotically approaches $\Delta t$ as $t$ goes to infinity for the observer, during which time the image is also infinitely dimmed and redshifted. If you tune the clock so that it ticks for the last time at exactly $\Delta t$, you never see that last tick in the reflection. Given arbitrarily sensitive instruments, the last image you measure reflected by the mirror (after a potentially very long wait, depending on how fast your clock ticks) is one tick before $\Delta t$.
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The clock will measure the proper time of an observer falling into the black hole. The observer will reach the horizon in a finite proper time; you will observe the clock reaching this value asymptotically (you will never see the clock actually reach the value it has when it reaches the horizon but you will see it get closer and closer).
The clock's light will also be redshifted as it approaches the horizon, meaning that the light will have longer and longer wavelengths (and smaller and smaller frequencies) until eventually you are not able to detect the light (assuming there's a maximum wavelength of light you can detect).
Andrew
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