I was watching a Japanese TV show, and in the start of the episode, introductions were made between two people; then later in the show, they were playing a video game. In both instances, the phrase "Hajimemashite" was used.
Is it correct to use it in the second instance? Shouldn't they have said, "Hajimemashou" instead?
Research
How do you do?; I am glad to meet you - はじめまして on Jisho
From my understanding, this is used when meeting someone for the first time.
to start; to begin; to commence; to initiate; to originate - 始めよう on Jisho
Whereas, this one means to begin, e.g., to start a race, game, etc.