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Let $d$ be a non zeo integer and $(x_1,...x_d)\in \mathbb R^d$

a) prove that for every $\epsilon$ , there exist an non zero integer $n$ and $(y_1,...,y_d)\in \mathbb Z^d$ such that for every $i\in\{1,...,d\}$ $|nx_i-y_i|<\epsilon$

b)Let's suppose that $(x_1,....x_d)$ is linearly independant in the $\mathbb Q$-vector space $\mathbb R$.show that for every $\epsilon$ >$0$ and for every $(a_1,...,a_d)\in \mathbb R^d$ ,there exist an non zero integer $n$ and $(y_1,...,y_d)\in \mathbb Z^d$ such that $i\in\{1,...,d\}$ $|nx_i-y_i-a_i|<\epsilon$

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    the first question seems easy , my idea for the second one is to complete the linearly indepent real number to a basis of R seen as $\mathbb Q$-vetor space and represent the $a_i$ in this basis , what do you think ? – 2017-01-19
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    It's unfair to close it as off-topic ?? – 2017-02-16

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