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This is the same problem as the seven bridges of Königsberg.
The simple explanation is:
Your path has one start point and one end point. Aside from the starting line and end line which arrive at these points, at every other corner/vertex you must both enter and exit, which requires an even number of edges/lines. For each of the 4 vertexes/corners, there are odd number of edges or connecting lines. Since there can only be 2 (or 0) odd numbered vertexes (for start and end), you are guaranteed to eventually land on a vertex which has no exit. So it can't be done.
If you add a second line across the top or bottom, you have made the number of vertexes even on the two connected corners, which makes it work
Is it possible to draw this shape without double lines or lifting your pen frame the paper? Or perhaps by only doubling one of the horizontal lines?