Does Monotone Convergence imply Convergence Subsequence?

[annoymage]

Homework Statement

Results

i) if (a_n) tends to L as n tends to infinity, then a_{n_r} tends to L as r tend to infinity

ii)if (a_n) tends to infinity as n tends to infinity, then a_{n_r} tends to infinity as r tend to infinity

using this result prove that

if (a_n) is an increasing sequence, prove that the converse of i) is true
Suppose (a_n) is divergent, then by ii), all the subsequences must be divergent, so, cant.

therefore (a_n) must be convergent, means (a_n) tends to M for some M,

apply i), then means M=L

but how come i didn't use the fact (a_n) is monotone, must be something wrong somewhere, help T_T

 

[fzero]

"Tends to infinity" is not the only way a sequence can diverge. Oscillating sequences are also divergent.

 

[annoymage]

aaaaaaaaaaaarghh, yes yes, thank you ^^

hmm, now i have to prove that <br /> (a_n)<br /> is convergent,

i suspect i should prove that <br /> (a_n)<br /> is bounded then, i know <br /> (a_n)<br /> is monotone then, <br /> (a_n)<br /> must be converging right? then continue like i was doing above right?

 

[fzero]

That sounds correct.

 

[annoymage]

thank you very much