Ceeb toom: nws muaj tseeb tias txhua qhov kev sib txuas ua ke muaj qhov sib txuas ua ke, thiab ntxiv mus, txhua qhov kev sib tw monotonic converges yog thiab tsuas yog bounded. Ntxiv Saib qhov nkag ntawm Monotone Convergence Theorem kom paub ntau ntxiv txog kev lav phib xaub ntawm bounded monotone sequences.
Puas txhua qhov kev sib txuas ua ke sib koom ua R?
Theorem hais tias txhua qhov sib luag hauv R muaj kev sib koom ua ke. Ib qho kev sib npaug yog qhov sib npaug ntawm R yog sib txuas ua ke yog thiab tsuas yog kaw thiab ciam teb. Lub theorem yog qee zaum hu ua qhov kev sib txuas sib txuas ua ke.
Puas yog txhua qhov sib lawv liag ntawm cov lej tiag tiag?
Teb thiab piav qhia: (a) Puas yog txhua qhov kev sib txuas ua ke sib txuas? No.
Puas yog txhua qhov kev cia siab ntawm monotonic sib txuas?
Tsis yog txhua qhov kev sib tw, zoo li (−1)n, converge, tab sis yog tias peb paub tias qhov kev sib tw bounded yog monotone, ces qhov no yuav hloov. yog ib qho ≥ ib + 1 rau tag nrho n ∈ N. Ib ntu yog monotone yog tias nws nce lossis txo. thiab bounded, ces nws converges.
Puas tag nrho cov kev sib txuas ua ke muaj kev sib koom ua ke?
The Bolzano-Weierstrass Theorem: Txhua qhov kev cia siab nyob hauv Rn muaj qhov sib txuas ua ke. ntawm {xmk } yog ib tug bounded ib theem zuj zus ntawm cov zauv tiag tiag, yog li nws kuj muaj ib tug convergent tom qab, … Hloov pauv, txhua qhov sib lawv liag yog nyob rau hauv ibkaw thiab bounded teem, yog li nws muaj ib tug convergent tom qab.
