\( \definecolor{colordef}{RGB}{249,49,84} \definecolor{colorprop}{RGB}{18,102,241} \)
Show by two different methods that the sequences \((1)_{n \in \mathbb{N}}\), \((n^2)_{n \in \mathbb{N}}\) and \((2^n)_{n \in \mathbb{N}}\) are linearly independent in \(\mathbb{R}^\mathbb{N}\).
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