Vector space/Countable basis/Flags/Exchange/Exercise

Let ${}V$ be a ${}K$-vector space, and let ${}v_{n}$, ${}n\in \mathbb {N} _{+}$, be a basis of ${}V$. Let ${}u_{n}$, ${}n\in \mathbb {N} _{+}$, be another family of vectors in ${}V$. Suppose that, for every ${}n\in \mathbb {N}$, the equality

${}\langle v_{1},\ldots ,v_{n}\rangle =\langle u_{1},\ldots ,u_{n}\rangle \,$

holds. Show that also ${}u_{n}$, ${}n\in \mathbb {N} _{+}$, is a basis of ${}V$.

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