Voadam
Legend
In TSR D&D that actually depends on the specifics of the edition and DM campaign set up choices.Could be, but wasn't always, depending on what spells the 5th-level mage could get access to. Remember, in the TSR editions you didn't get to choose your spells; what you got at level-up was random and after that it was all about what you happened to stumble across and-or what other mages in the party happened to have and were willing to let you learn.
Moldvay Basic:
"Each magic-user and elf has a spell book for the spells that he or she has learned. A first level character will only have one spell (a
first level spell) in the spell book. A second level character will have two spells (both first level) in the spell book; a third level character will have three spells (two first level spells and one second level spell) in the spell book. The DM may choose which spells a character has in the book, or may allow the player to select them."
2e PH:
Whenever a specialist reaches a new level, he automatically gains one spell of his school to add to his spell books. This spell can be selected by the DM or he can allow the player to pick. No roll for learning the spell need be made. It is assumed that the character has discovered this new spell during the course of his research and study.
The closest I can find on OD&D explaining new spell acquisitions is "Characters who employ spells are assumed to acquire books containing the spells they can use, one book for each level. If a duplicate set of such books is desired, the cost will be the same as the initial investment for research as listed above, i.e. 2,000, 4,000, 8,000, etc. Loss of these books will require replacement at the above expense."