Prove that $\{2^n : n \in \mathbb{N} \}$ is not bounded.
Proof: By Induction; $2^n \geq n,\forall n \geq 1$.
By Archimedean property $\mathbb{N}$ is not bounded above.
Assume that $\{2^n : n \in \mathbb{N} \}$ is bounded.
Then $\exists M \in \mathbb{R}\gt 0$
such that $\forall n, n \leq 2^n \leq M$ $ \Rightarrow \mathbb{N} $ is bounded . Contradiction.
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