abstract algebra - Linear independence over $mathbb Q$ and $mathbb R$ of subsets of $2^{mathbb N}$

I have the following doubt:

Suppose $f_1,\ldots,f_n\in 2^{\mathbb N}$ are such that $\{f_1,\ldots,f_n\}$ is linearly independent in the $\mathbb Q$-vector space $\mathbb{Q^N}$. Does this set remain linearly independent in the $\mathbb R$-vector space $\mathbb{R^N}$?

Here $2=\{0,1\}$. I would like hints, not full answers.

Thanks

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Edit: I have shown that if there is some $I\subseteq\Bbb N$ such that $f_1\upharpoonright I,\ldots,f_n\upharpoonright I$ is linearly independent over $\Bbb Q$ with $|I|\geq n$,then we are done, however I can't see why such $I$ should exist.

Answer

Suppose $\lambda_1 f_1 + \cdots + \lambda_n f_n = 0$, where $\lambda_1,\dots,\lambda_n\in\mathbb{R}$. Try picking a basis for the $\mathbb{Q}$-vector space spanned by $\lambda_1,\dots,\lambda_n$.