I have a question concerning black holes and hawking radiation. I hope it is an adequate question and I pose it in the right section.

Consider a free-falling observer 1 approaching the event horizon of a black hole. Evaluating his proper time, one finds that he crosses the event horizon within finite time. For an asymptotic observer 2 however, the time it takes observer 1 to cross the event horizon is infinite. However, the time it takes for the black hole to evaporate due to Hawking radiation is finite. So observer 2 would see the black hole evaporate before observer 1 reaches the event horizon. Is that correct? And what would observer 1 see? The Universe, including the black hole accelerating in time so that the black hole vanishes in radiation before he can reach it?