Main Question or Discussion Point
Suppose an observer in the earth frame set out to measure the length AB by measuring the time interval between the event "B coincides with D" (event BD) and the event "A coincides with D" (event AD), and then multiplying the time with ##v##. This observer argues that an observer in the train frame would measure this time to be smaller since moving clocks run slower, and so would arrive at a shorter length for AB. But to the observer in the train frame, length AB is stationary, a proper length, and so should instead be longer. We arrive at a contradiction.
Where's the mistake?
Note: The train observer measures the time interval between event BD and event AD and then multiply it by ##v## to get the length AB. In the train frame, the Earth and hence point D are moving at speed ##v## too. The distance traveled by D during this time is the length AB.