proving limit representation of e

[kamil9876]

According to my lecturer, this one was in last year's exam.

Given that

Where n is a natural number.

Prove that

Where x is any positive number, not neccesarily an integer.

-----------------------------------------------------------------------------

Now our lecturer mentioned that sandwhich theorem is the way to go.

n<x<n+1 implies:



And by taking limit n approaches infinity of everything u get the upper bound=lower bound= e.

Now here is my issue:
I do agree that upper bound=e since this can be proved by algebraic manipulation:



And by the multiplication property of limits and the given statement this is proved.

But how do u prove if for the lower bound? I'm after somethign rigorous because I am not convined by the argument: as n-->infinity, n+1---> infinity, aka   since this is true if the function is monotonic, but not neccesarily if it is not. Also this is not exactly the case for the lower bound because there is an n+1 in the fraction, and n in the exponent.

Thanks in advance.

[kamil9876]

dw guys. I worked it out.

Lower bound can be represented as