Assume that U is a multiple of V. Then for some coefficient, c in set of reals, U=cV. Consider the distinct vectors U,V and distinct, real scalars a, b. If possible, we want to find a and b, nonzero, such that aU + bV=0 By substitution, U=cV and the sum can be written as a(cV)+bV. c is nonzero by definition as well as a,b. Let us choose b such that ac = -b, which is clearly real and all scalars remain distinct. The sum now becomes with substitution , (ac)V+bV -> (-b)V+bV which is clearly zero. This representation in non-trivial by definition and proves that U and V are linearly dependent.
Something like that.