Proof by induction (how did I do?)

Hi everyone, I'm new to the forums. I'm also fairly new to proofs, and I still commonly make mistakes, specifically regarding quantification errors and circularity. However, I believe that I have gotten this particular proof nailed down pretty well. I was wondering if I could get some feedback on whether or not what I did seems correct. I'm wondering specifically about the fourth line of the induction part, where I said "Note that this equation may also be written as [blah]." Should I elaborate on why, or is it obvious enough?

(Also, sorry about the alignment of some of the longer equations, I can't seem to get it to work like I normally do).

Any help is greatly appreciated!

**Theorem**: For all

**Proof**: We will proceed by induction.

i) Base step: Let . We see that . We also see that

Therefore, is true.

ii) Inductive step: Let . Assume that . We would like to show that . Note that this equation may also be written as . By our assumption, we see that

Therefore, is true.

Furthermore, by the PMI, is true for all .