平面上兩點(diǎn)x,y的距離記為D(x,y).
由d = sup{D(x,y) | x,y∈E},存在E中點(diǎn)列{x[n]}與{y[n]},使d-1/n < D(x[n],y[n]) ≤ d.
E是有界閉集,故點(diǎn)列{x[n]}存在收斂子列{x[n[k]]},收斂于某點(diǎn)a∈E.
設(shè)z[k] = x[n[k]],w[k] = y[n[k]].
則由n[k] ≥ k,d-1/k ≤ d-1/n[k] < D(x[n[k]],y[n[k]]) = D(z[k],w[k]) ≤ d.
再由E是有界閉集,點(diǎn)列{w[k]}存在收斂子列{w[k[i]]},收斂于某點(diǎn)b∈E.
設(shè)u[i] = z[k[i]],v[i] = w[k[i]].
則由k[i] ≥ i,d-1/i ≤ d-1/k[i] < D(z[k[i]],w[k[i]]) = D(u[i],v[i]) ≤ d.
在上式中令i → ∞,有D(u[i],v[i]) → d.
由u[i]是z[k]的子列,z[k]收斂到a,有D(u[i],a) → 0.
又v[i]收斂到b,有D(v[i],b) → 0.
而由三角不等式,D(a,b) ≥ D(u[i],v[i])-D(u[i],a)-D(v[i],b).
令i → ∞即得D(a,b) ≥ d.
但a,b∈E,由d = sup{D(x,y) | x,y∈E},得D(a,b) ≤ d.
故D(a,b) = d,a,b即為滿足要求的點(diǎn).