Annihilator proof

T

The blackjet

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Can someone please tell me if this is a correct proof?
Thank you
(proof in image)
 

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Welcome to FMH!

I found it quite hard to read your proof by induction.

I can follow your logic for the case n=2. Except I did not understand your notation D(xx) for "second derivative". Maybe you meant D2(xemx)? You made what looks like a copying out mistake on that same line writing x2 instead of m2 (but on the next line you correct this).

I think that your extension to "general n" is OK. But I have never seen this notation where an operator can be multiplied by itself before, so I can't be sure.

I would state that you have only proved it true for n≥2 (I think it may also hold for n=1 but you have not proved this?)
 
The blackjet said:
thank you for replying
So
D x D is not equal to D^2 as in second differential
Click to expand...

It could be correct. I have never seen mathematics like that before, therefore I can't say yes or no. Sorry.

But everything else I did understand, and I agreed with it. So I think you did well.

I found it hard to read because...
  • I am accustomed to writing the symbol "x" differently. I write more curved like \(\displaystyle x\). Straight x looks like a multiply symbol to me. I think different countries have different standards for this - so this might be correct for where you are currently learning.
  • I would write a few words to say what is happening in the algebra. "Prove by induction" -> "Prove for the case n=2" ... " LHS= ... =RHS therefore the statement is true for n=2"
  • The final part of your proof, "Assume true for n=k-1 and try to prove for the case n=k" was not entirely clear to me BUT I think I could see your logic. It was not presented very clearly in my opinion.
I am interested, what course is this from?
 
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