- How did Lyman discover his series in hydrogen atom?
- How did he know that the final energy level is the first level and not the second or the third or etc.?
- Or how did the other scientists know which level the electron is moving from?
He was directed into spectroscopy by his advisor. At the time the equipment was pretty poor for spectral measurements and much of his time he spent trying to get good spectral wavelength measurements. Part of the measurement error ended up coining the term "Lyman ghosts" in the spectral lines due to periodic errors of long period in the ruling of the grating. This led him to his PhD topic. He was very methodical and careful in his testing. It took many years for the process to become good enough so that in 1914 following the atomic model presented by Bohr and the Paschen Series for Hydrogen (observed in 1908) he began to observe and test his data against the idea of what is now the Lyman series. In 1914 he announced the first 3 in the series that fit the new models.
Most of this early work was based on observations and trying to fit the Bohr model. Hard to say what he was thinking exactly but he was very careful in his testing and analysis to make sure instrument and test errors were not a factor. This work is probably what led him to the right track in knowing he was observing forms of hydrogen and how they matched the model.
I found a memoir of him on line, which you might like to read.