Geometry provides insight and classical inequalities afford easy access to rigor.
Geometric solution
We know, from the geometry of least squares, that x¯=(x¯,x¯,…,x¯) is the orthogonal projection of the vector of data x=(x1,x2,…,xn) onto the linear subspace generated by the constant vector (1,1,…,1) and that σx is directly proportional to the (Euclidean) distance between x and x¯. The non-negativity constraints are linear and distance is a convex function, whence the extremes of distance must be attained at the edges of the cone determined by the constraints. This cone is the positive orthant in Rn and its edges are the coordinate axes, whence it immediately follows that all but one of the xi must be zero at the maximum distances. For such a set of data, a direct (simple) calculation shows σx/x¯=n−−√.
Solution exploiting classical inequalities
σx/x¯ is optimized simultaneously with any monotonic transformation thereof. In light of this, let's maximize
x21+x22+…+x2n(x1+x2+…+xn)2=1n(n−1n(σxx¯)2+1)=f(σxx¯).
(The formula for f may look mysterious until you realize it just records the steps one would take in algebraically manipulating σx/x¯ to get it into a simple looking form, which is the left hand side.)
An easy way begins with Holder's Inequality,
x21+x22+…+x2n≤(x1+x2+…+xn)max({xi}).
(This needs no special proof in this simple context: merely replace one factor of each term x2i=xi×xi by the maximum component max({xi}): obviously the sum of squares will not decrease. Factoring out the common term max({xi}) yields the right hand side of the inequality.)
Because the xi are not all 0 (that would leave σx/x¯ undefined), division by the square of their sum is valid and gives the equivalent inequality
x21+x22+…+x2n(x1+x2+…+xn)2≤max({xi})x1+x2+…+xn.
Because the denominator cannot be less than the numerator (which itself is just one of the terms in the denominator), the right hand side is dominated by the value 1, which is achieved only when all but one of the xi equal 0. Whence
σxx¯≤f−1(1)=(1×(n−1))nn−1−−−−−−−−−−−−−−−√=n−−√.
Alternative approach
Because the xi are nonnegative and cannot sum to 0, the values p(i)=xi/(x1+x2+…+xn) determine a probability distribution F on {1,2,…,n}. Writing s for the sum of the xi, we recognize
x21+x22+…+x2n(x1+x2+…+xn)2=x21+x22+…+x2ns2=(x1s)(x1s)+(x2s)(x2s)+…+(xns)(xns)=p1p1+p2p2+…+pnpn=EF[p].
The axiomatic fact that no probability can exceed 1 implies this expectation cannot exceed 1, either, but it's easy to make it equal to 1 by setting all but one of the pi equal to 0 and therefore exactly one of the xi is nonzero. Compute the coefficient of variation as in the last line of the geometric solution above.