Theorem: Every convergent sequence is bounded.
Proof: Let $\{a_n\} \rightarrow a$
Choose $\epsilon = 1$, and by the definition of convergence there exists $N$ such that $$|a_n - a| < 1$$ for all indices $n \geq N$ Let's start from a tautology: $$|a_n| = |a_n - (a - a)|$$ Rearrange $$|a_n| = |(a_n - a) + a|$$ Use the Triangle inequality $$|a_n| \leq |a_n-a| + |a|$$ Now we use our inequality from the definition of convergence $$|a_n| \leq 1 + |a|$$ for all indices $n \geq N$. Let $$M = \max(1+|a|,|a_1|,|a_2|,...,|a_{N-1}|)$$ Then $$|a_n| \leq M$$ for every index n, which satisfies the definition of boundedness. Therefore $\{a_n\}$ is bounded.