How to prove $u = \mathrm{span}\{v_{1},\dots,v_{n}\}$ The vectors $v_{1},\dots,v_{n}$ form a linearly independent set.
Addition of vector $u$ makes this set linearly dependent.
How to prove that $u \in span\{v_{1}\ldots,v_{n}\}$.
I was able to prove the reverse.

There are $a_0, \dots, a_n$ not all zero, such that
$$a_0u + a_1v_1 + \dots a_nv_n = 0.$$
If $a_0 = 0$, the set will become independent, which is contradictory.
Clearly $a_0 \neq 0$, as $v_1, \dots, v_n$ are linearly independent, and therefore we get
$$u = -\frac{a_1}{a_0}u_1 - \dots - \frac{a_n}{a_0}u_n $$
