å¹³é¢ä¸ä¸¤ç¹x,yçè·ç¦»è®°ä¸ºD(x,y).
ç±d = sup{D(x,y) | x,yâE},åå¨Eä¸ç¹å{x[n]}ä¸{y[n]},使d-1/n < D(x[n],y[n]) ⤠d.
Eæ¯æçéé,æ
ç¹å{x[n]}åå¨æ¶æåå{x[n[k]]},æ¶æäºæç¹aâE.
设z[k] = x[n[k]],w[k] = y[n[k]].
åç±n[k] ⥠k,d-1/k ⤠d-1/n[k] < D(x[n[k]],y[n[k]]) = D(z[k],w[k]) ⤠d.
åç±Eæ¯æçéé,ç¹å{w[k]}åå¨æ¶æåå{w[k[i]]},æ¶æäºæç¹bâE.
设u[i] = z[k[i]],v[i] = w[k[i]].
åç±k[i] ⥠i,d-1/i ⤠d-1/k[i] < D(z[k[i]],w[k[i]]) = D(u[i],v[i]) ⤠d.
å¨ä¸å¼ä¸ä»¤i â â,æD(u[i],v[i]) â d.
ç±u[i]æ¯z[k]çåå,z[k]æ¶æå°a,æD(u[i],a) â 0.
åv[i]æ¶æå°b,æD(v[i],b) â 0.
èç±ä¸è§ä¸çå¼,D(a,b) ⥠D(u[i],v[i])-D(u[i],a)-D(v[i],b).
令i â âå³å¾D(a,b) ⥠d.
ä½a,bâE,ç±d = sup{D(x,y) | x,yâE},å¾D(a,b) ⤠d.
æ
D(a,b) = d,a,bå³ä¸ºæ»¡è¶³è¦æ±çç¹.
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