Properties of valuation map

Question

It seems not so hard to prove but how can we prove by induction. Let $K$ be a field and $\nu$ be a valuation map. If $a_{1} + a_{2} + ... + a_{n} = 0$ then prove that $\nu(a_{i}) = \nu(a_{j})$ for some $i \neq j$, where $a_{1}, a_{2},...,a_{n} \in K$. Thanks

Answer 1

Prove by induction on $n$ the following: if $v(a_1)>\cdots >v(a_n)$, then $v(a_1+\cdots+a_n)=v(a_n)$.