Prove the inequality.

[Harsh123]

[math]\text { 35. Let, } x_{n}=\sqrt[2]{2+\sqrt[3]{3+\sqrt[4]{4+\cdots+\sqrt[n]{n}}}} ; \text { prove that } x_{n+1}-x_{n}<\frac{1}{n !}, n=2,3, \ldots .[/math]

 

[Zermelo]

We can’t help you unless you show some of your work. If I were you, I would try mathematical induction.

 

[Harsh123]

We can’t help you unless you show some of your work. If I were you, I would try mathematical induction.

But how can we do it by induction?

But how can we do it by induction?

I really don't know how to initiate the solution

 

[Deleted member 4993]

But how can we do it by induction?


I really don't know how to initiate the solution

Go to Google.com and do a search with following keywords:

Proof by induction​

You'll find many videos/papers - ready to teach you. Study some of those - and those will give you some idea about how to initiate the solution.

If you do not understand any particular point in those "lesson videos/papers" please come back and ask about it

 

[Dr.Peterson]

But how can we do it by induction?


I really don't know how to initiate the solution

Please tell us the context of the problem, and what you DO know. Presumably, if you were given this problem to solve, you have some knowledge (or at least someone has tried to provide you with that knowledge) with which to solve it. Possibly knowing that would give us more ideas besides induction to suggest.

We are here to help you, not just to give answers or to teach you from scratch. So far, you've shown no evidence of effort.