My course notes (Mathematics BSc, second year module in real analysis, unpublished) have,
Theorem 2.1.8 If a sequence $(a_n)$ is convergent, then it is bounded.
The proof includes,
[...] now suppose that $n \leq N$. We then have $$|a_n|\leq\text{max}\{|a_1|,|a_2|,\dots,|a_N|\}.$$
But doesn't assuming the existence of a maximum assume boundedness, the thing to be proved? I'm guessing there's a tacit assumption that "infinities" can only exist at the "end" of a sequence of infinite length, not as singularities within a sequence of finite length. Am I right?