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 $(a_n)$ is convergent,

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

 

[fzero]

That sounds correct.

 

[annoymage]

thank you very much