Classic linear algebra exercise: the union of a subspace is a subspace if and only if one is contained in the other. This is also good practice with the definition of a subspace, and also shows how to prove statements of the form p implies (q or r)
Check out my vector space playlist: https://www.youtube.com/playlist?list=PLJb1qAQIrmmDhpRvkRXUMNGfV3LVZwL_w
Subscribe to my channel: youtube.com/c/drpeyam