Proof
(related to Proposition: Complex Convergent Sequences are Bounded)
 By hypothesis, \((a_n)_{n\in\mathbb N}\) is a convergent complex sequence with \(\lim_{n\rightarrow\infty} a_n=a\).
 Thus, there is an \(N\in\mathbb N\) such that $a_n  a < 1\quad\quad\forall n\ge N.$
 By virtue of the triangle inequality, we get $$a_n = a_n  a + a\le a_n  a + a < 1 + a$$ for all $n\ge N.$
 Set \(B:=\max(a_0,a_1,\ldots,a_{N1},1 + a)\).
 With this constant, we have $a_n \le B$ for all $n\in\mathbb N.$
 Therefore, \((a_n)_{n\in\mathbb N}\) is bounded.
∎
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References
Bibliography
 Kane, Jonathan: "Writing Proofs in Analysis", Springer, 2016