[閒聊] proof

作者winston628 (阿搞( ￣ c￣)y▅▅ξ)

看板Megan

標題[閒聊] proof

時間Tue Apr 21 01:32:06 2009

A nondecreasing sequence which is bounded above converge to the least upper bound of its range pf:Let{an}n=1~infinite be the sequence as above so a1<=a2<=a3<=.... let S={an|1<=n<=infinite} and M=the least upper bound Then we are going to show lim an = M that for epsilon>0 exist k belong to N such that for all n>=k |an-M| <epsilon first since M is an upper bound for all n we have an<=M so for epsilon>0 we have an<M+ epsilon for all n belong to N On the other hand for epsilon >0 by them11.1.2 exist k belong to N such that M-epsilon<ak moreover sine{an} is a nondecreasing sequence M-e<ak<=an n>=k Hence for all n>=k M-e<an so for this epsilon >0 and thos h belong to N we have M-e<an<M+e n>=k Hence we have |an-M|<epsilon ----------------------------------------- Prouduct rule lim an = L , lim bn = M , lim an*bn=L*M pf: for all epsilon>0 exist k1 belong to N n>=k1 such that |an-L|<e/2(|M|-1) for all epsilon>0 exist k2 belont to N n>=k2 such that |an-L|<1 an<|L|+1 for all epsilon>0 exist k3 belong to N n>=k3 such that |bn-M|<e/2(|L|-1) take k=Max{k1,k2,k3} |anbn-LM|=|anbn-anM+anM-LM|<=|an||bn-M|+|M||an-L| <|L+1||bn-M|+|M+1||an-L|=e/2+e/2=e -- ※ 發信站: 批踢踢實業坊(ptt.cc) ◆ From: 140.114.218.19

推 heyheychi :全英文的証明欸！菸小弟你不錯噢(拍) 04/21 01:33

→ winston628 :XDD 04/21 01:33

推 sponge0823 :爽 04/21 01:34

→ sponge0823 :32分還是要水桶XD 04/21 01:34

→ winston628 :不早說我在推文那浪費了太多時間不然應該來得及 04/21 01:35

→ winston628 :老娘看到這個會不會以為走錯版 04/21 01:35

→ WOWARTHUR777:阿搞的證明我看不懂.... (畫圈) 04/21 01:35

→ winston628 :XDDD 04/21 01:35

→ moonstars :小菸是用背的嗎? 04/21 01:37

→ winston628 :是的 04/21 01:37

推 sponge0823 :說真的你背了出社會也用不到 04/21 01:37

→ winston628 :被這個只是為了應付考試= = 不然這個敘述很基本= = 04/21 01:38

→ moonstars :可是要先應付考試..說不定小菸之後當老師 04/21 01:38

→ moonstars :這樣就會用的到..。XD 04/21 01:38

→ winston628 :拒絕當老師XDDD 04/21 01:38

→ hohoholalala:我要給小菸一個章!! 04/21 01:39

→ moonstars :不然海綿其實我們都可以不用上學阿..在家被爸媽教 04/21 01:39

→ mishka :這篇是天書 04/21 01:39

→ moonstars :一加一等於二就夠用了這樣 04/21 01:39

→ winston628 :XDD 有計算機就夠了這樣XDD 04/21 01:40

推 sponge0823 :XDDDDDDDDDDDDDDDDDDDDDDDDDDDDD 04/21 01:41

→ moonstars :黑呀..不過受教育應該還有更多東西要學，例:團體生活 04/21 01:42

→ winston628 :EX:宅宿舍 04/21 01:42

→ moonstars :小菸只考一題嗎?你應該要繼續在推文中默寫?! 04/21 01:42

→ winston628 := = 04/21 01:43

※ 編輯: winston628 來自: 140.114.218.19 (04/21 01:50)

→ mishka :XDDDDDDDDDDDDDDDDDDDDDDDD 04/21 01:45

→ moonstars :XDDDDD 我是讓你享有推文的樂趣兼顧念書耶...讚! 04/21 01:45

推 moonstars :XD 優 04/21 01:52

→ moonstars :老娘會不會有一天也把論文寫在這呢?(期待) 04/21 01:52

→ winston628 :XDDD 釣魚文? 顆顆 04/21 01:53

※ 編輯: winston628 來自: 140.114.218.19 (04/21 01:56)

→ moonstars :XDDDDDD 拋磚引玉這樣..。XD 04/21 01:58

推 sjlluvia :越來越多了耶XDD 04/21 02:03

→ WOWARTHUR777:一樣又是我看不懂的證明 (畫圈) 04/21 02:04

推 moonstars :包子我也看不懂(握) 04/21 02:12

→ moonstars :等小菸考完再叫他解釋一下(翻成中文這樣) 04/21 02:13

推 sjlluvia :我也看不懂(握握) 04/21 02:13

推 WOWARTHUR777:那我們看不懂得去旁邊坐一桌這樣看會不會有第四咖 04/21 02:15

→ sponge0823 :要打麻將了嗎 04/21 02:16

推 moonstars :海綿要來湊嗎?(可是我不會打會一直放槍喔) 04/21 02:21

推 sponge0823 :我也差不多啦可是又很愛打 04/21 02:22

推 sjlluvia :我也不會打...... 04/21 02:22

推 WOWARTHUR777:哈~~我也不會耶!可是我會看一點點 04/21 02:23

→ WOWARTHUR777:所以...我們四個要湊一桌打嗎? 04/21 02:23

推 moonstars :XD 所以整桌都在放槍然後一下子就胡了這樣嗎? 04/21 02:38

推 WOWARTHUR777:我覺得我們有空可以一起玩一下!看誰放槍的多~XDD 04/21 02:39

推 moonstars :海綿說板聚要打麻將這樣...XD 04/21 02:41

推 sjlluvia :所以下次版聚是麻將放槍大賽嗎XDDD 04/21 02:42

→ moonstars :應該是喔..阿咧小米還在這樣..XD 04/21 02:42

推 sjlluvia :哈哈還在啊我打作業都會一直掛著PTT 04/21 02:44

→ sjlluvia :然後不時跑來看一下XD 04/21 02:44

推 WOWARTHUR777:喔喔~~~所以下次版聚就是麻將大賽~一桌比放槍.一桌比 04/21 02:46

→ WOWARTHUR777:贏錢這樣?XDDD 04/21 02:46

推 sjlluvia :那地點咧...還是在哈比屯嗎XDD 04/21 02:47

→ moonstars :XDDDDD 我以為小米去睡了...原來還在趕報告 04/21 02:49

→ moonstars :地點? 問海綿好了..海綿咧? 04/21 02:49

推 WOWARTHUR777:然後還會有一桌是在偷偷觀察打牌的那兩桌這樣~XDDD 04/21 02:51

→ winston628 :=====================3:00================ 04/21 03:01

推 WOWARTHUR777:阿搞~你還沒睡阿? 04/21 03:03

→ etlu :我走錯版了嗎 XD 04/21 03:07

推 WOWARTHUR777:沒阿!白白~~這裡是老娘麻將版這樣! 04/21 03:08

→ WOWARTHUR777:XDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD 04/21 03:08

→ winston628 :我還在讀書阿 04/21 03:09

推 WOWARTHUR777:阿搞好認真喔!!!加油喔!!! 04/21 03:09

推 etlu :這一篇完全看不懂是怎樣 XDDDD 04/21 03:13

→ winston628 :海綿威脅我打的 04/21 03:14

推 moonstars :哈哈..。是我起的頭..拍謝 04/21 03:19

→ moonstars :晚點名第14頁有 04/21 03:19

→ winston628 :這樣背比較熟XDD 明天大家看到一定會一樣的反應XDD 04/21 03:22

推 sjlluvia :噢我解決掉一篇報告了(灑花) 04/21 03:23

推 etlu :好好笑 XD 是什麼的證明啊感覺我的統計有可愛很多 04/21 03:25

→ winston628 :級數收斂的無聊證明!!!!!!!!!! 04/21 03:32

推 WOWARTHUR777:話說...大家在仙劍的哪一區哪個伺服器阿?(狀況外) 04/21 03:35

推 sjlluvia :我還沒開始玩仙劍OL耶...看到3D我就暈了一半ˊˋ 04/21 03:38

→ winston628 :似乎是逍遙? 04/21 03:38

推 WOWARTHUR777:埃...結果剛剛要玩阿...卻告訴我...顯示卡過舊...T^T 04/21 03:40

→ WOWARTHUR777:好想哭....我用很久耶!!!(／‵Д′)／~ ╧╧ 04/21 03:41

→ winston628 :<( ￣︶￣)╱(._. ) 有點暈了來去睡覺囉晚安 04/21 03:47

推 WOWARTHUR777:阿搞~~估奈!! 04/21 03:49

推 moonstars :包子還不睡?! 04/21 04:19

→ moonstars :XDDDD 我如果當老師一定很好笑... 04/21 04:20

推 sjlluvia :月星還在喔~ 先說聲晚安我要去躺平了:) 04/21 04:24

推 moonstars :小米晚安! 04/21 04:25

→ etlu :大家晚安 04/21 04:53

→ etlu :好煩英文作業寫的我詞窮好多summary要寫 (哭哭) 04/21 04:54

推 moonstars :早安! 04/21 09:38

推 sponge0823 :早安 04/21 10:59

推 mishka :早安~~~XD 04/21 11:12

推 sponge0823 :你那邊幾點?XDDD 04/21 11:12

→ mishka :我昨天跟小兒聊MSN聊到睏去XD 就一覺到剛剛XD 04/21 11:13

→ mishka :我這邊早上七點13分XD 04/21 11:13

推 sponge0823 :好遭= = 04/21 11:23

推 mishka :XDDDDDDD 04/21 11:28

→ winston628 :考完了!!!!!!!!!!!!!!!!!!!! 04/21 11:32

推 mishka :恭喜小菸~~~~不過我要去睡回籠覺了超累 = = 04/21 11:35

→ winston628 :XDDD 晚安~~ 04/21 11:36

推 sponge0823 :晚安XD 04/21 11:37

→ winston628 :http://tinyurl.com/co2khz 這是考卷XDDD 04/21 11:39

→ winston628 :上面的證明是第一跟第二題 04/21 11:39

推 sponge0823 :XD 04/21 11:43

推 moonstars :還可以拍起來喔? 04/21 12:23

推 peijou :老師!!小菸帶手機...作弊啦!(舉手) 好餓喔 04/21 12:26

推 eakw :偷打廣告 Tender版有奇怪的投票進行中 (飄) 04/21 12:45

推 peijou :投完了!!YES再10分鐘就可以下課去吃飯了!! 04/21 12:50

→ winston628 :考卷可以帶回來~ 04/21 12:51

推 rene611 :午安~大家~ 04/21 13:08

→ rene611 :昨天依整爆忙...沒時間來~@@ 04/21 13:09

→ winston628 :剛安裝玩仙劍申請完帳號明天還有考試XDD 考完再玩 04/21 14:20

推 WOWARTHUR777:我起床了....還想睡....= = 04/21 14:44

→ Onuris :這……… 04/21 14:52

推 mishka :我睡醒了 XD 04/21 15:54

推 fastgoat :等會要去上課了~~今天很早就要去上課了 04/21 16:04

推 mishka :都沒有人XDDDDD 04/21 17:16

推 hwaji :我又病了這樣XD 04/21 17:54

→ hwaji :台南大學放榜了喔XDDDDDD 04/21 17:55

→ hwaji :翔翔上了沒?XD 04/21 17:56

→ winston628 :某郎 04/21 18:27

推 moonstars :去查一下榜.. 04/21 18:28

→ moonstars :晚點名咧?我今天忙了一整天這樣= = 04/21 18:29

推 mishka :今天都一直沒人欸XD 04/21 18:29

→ winston628 :XDD 都跑去玩仙劍喔 04/21 18:31

推 mishka :似乎是這樣敲海綿完全沒反應這樣XD 04/21 18:33

推 moonstars :翔翔備取生耶http://0rz.tw/fIW5u 04/21 18:33

→ moonstars :我剛剛去查了這樣...XDDDDD 04/21 18:33

→ winston628 :我在線上有看到他傅咩XDDD 04/21 18:33

→ mishka :他掛忙碌啊水球也砸不醒XDDDDD 04/21 18:34

推 moonstars :是喔...哈哈哈哈 04/21 18:35

推 WOWARTHUR777:海綿已經完全中仙劍毒這樣!好好喔~我要也玩啦!! 04/21 18:35

→ moonstars :正取20名，備取8名 04/21 18:36

→ moonstars :噗!打電話給他...XD 04/21 18:36

→ WOWARTHUR777:大家都來砸海綿水球~~讓她醒一下!XDDDD 04/21 18:36

→ moonstars :水球應該沒用吧.. 04/21 18:37

→ moonstars :包子開晚點名啦! 04/21 18:37

推 WOWARTHUR777:可是我不知道要喇啥~所以啦!月星你開~(推) 04/21 18:39

→ winston628 :XDDDDDDDDD 04/21 18:39

→ WOWARTHUR777:阿搞笑什麼?(搔頭) 04/21 18:39

→ winston628 :沒什麼顆顆 04/21 18:40

推 moonstars :喇鴨舌這樣XDDD 04/21 18:40

→ WOWARTHUR777:說一下嘛...(推) 04/21 18:40

→ winston628 :傻笑顆顆 04/21 18:42

推 moonstars :米西卡咧?該不會在寫了吧?(歪頭) 04/21 18:43

推 WOWARTHUR777:乾脆就直接這篇當點名文就好啦!等它爆了再開~(茶) 04/21 18:44

→ winston628 :可能哦XDDD 這樣聊著聊著這篇也快爆了XDD 04/21 18:44

推 moonstars :月底又到了我又要開始大忙了這樣 04/21 18:47

→ winston628 :月底又到了我又要開始拮据了這樣 04/21 18:48

→ winston628 :某郎挖馬來造 04/21 18:54

推 WOWARTHUR777:阿搞!!!(抓) 04/21 18:56

→ winston628 :XDD 挖要來去忙了估掰晚點見 04/21 18:57

推 mishka :我在吃午餐這樣XD 04/21 18:58

推 WOWARTHUR777:阿搞晚點見阿!米西卡中午吃啥?(好奇貌) 04/21 18:59

推 moonstars :包子快點po晚點名啦..(推)我手上還有三份圖沒畫..= = 04/21 19:07

推 lovemasami :冏爆老娘連三天都有上來都沒遇到這樣 XDDDD 04/21 19:10

推 rene611 :我還沒下班@@ 04/21 19:14

→ rene611 :最好是找到新人~還越忙越晚這樣~ 04/21 19:16

推 WOWARTHUR777:好啦!我有PO了...請各位移駕吧~(茶) 04/21 19:18

→ mishka :包子我午餐吃麥當當這樣...... 04/21 20:10

→ peijou :俄國的麥當當好吃嗎?? 04/21 20:11

推 mishka :跟台灣一樣啊XDDDDD 04/21 20:12

推 sponge0823 :XDDDDDDDDDDDDDDDDD 04/21 20:13

→ peijou :聽說每一個國家都有特別的口味...原來是假的! 04/21 20:14

推 mishka :喔他最近出了墨西哥雞肉捲結果裡面是麥香雞的雞肉 04/21 20:25

→ mishka :不是很好吃這樣.... 04/21 20:26

→ peijou :麥香雞的雞肉 (冏)XDDD 04/21 23:40

→ winston628 :按進來這篇真的有走錯版的FU XDDDD 04/21 23:57

推 wenneyy :XDDDDDDDDDDDDDDDDDDDDDDD 麥香雞的雞肉 XDDDDDDDDDD 04/22 00:06