Understanding Induction: Solving a Challenging Step

[transgalactic]

i can't understand this step in this induction

http://img383.imageshack.us/img383/5677/99884212ia5.gif

 

[tiny-tim]

Because if an + L - e < an+1,

then an + 2(L - e) < an+1 + L - e

but an+1 + L - e < an+2

so an + 2(L - e) < an+1 + L - e < an+2

Now try it for the other half, and write it out properly as an induction proof.

 

[transgalactic]

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

 

[transgalactic]

this is the expanded proove
http://img220.imageshack.us/img220/1912/13164747xs4.gif