i cant understand this step in this induction
http://img383.imageshack.us/img383/5677/99884212ia5.gif
i cant understand this step in this induction
http://img383.imageshack.us/img383/5677/99884212ia5.gif
It would have been clearer if you had posted the whole argument. This is part of a proof by induction of what? Actually, it looks like the result of induction.
You know that for every n, $\displaystyle a_n- L- \epsilon< a_{n+1}< an+ L+ \epsilon$ so $\displaystyle a_n- k(L+ \epsilon)< a_{n+k}< a_n+ k(L+ \epsilon)$ is certainly true for k= 1. Now suppose $\displaystyle a_n- k(L+ \epsilon)< a_n< a_{n+1}+ k(L+ \epsilon)$ is true for some specific k and all n. Then a_n- (k+1)(L+ \epsilon)= [a_n- (L+ \epsilon)]- k(L+\epsilon)< a_{n+1}-k(L+\epsilon)[/tex] and now use [tex]a_n- k(L+ \epsilon)< a_{n+1}[/itex] with n+1 nstead of n- which you can do because it is true for all n.
this is the expanded proove
http://img220.imageshack.us/img220/1912/13164747xs4.gif
in the total case of K
we are given some expression
and we presume that its true
so to prove the K+1 case
from where did they use the left side development
it seems that the are using the given expression itself to proove it
?
http://img149.imageshack.us/img149/6359/10189365rk3.gif