proof that {(1+x/n)^n} is bounded

Hello,

how can I prove that the set E(x) = { (1+x/n)^n} is bounded ? (n is a natural number and x is a positive real number)

I had already proven that the set has no largest element using a consequence of the AGM inequality.

I also know from calculus that the limit of (1+x/n)^n is actually e^x . but this is of no use to me, since e^x is not yet defined in my analysis course.

any help would be much appreciated.

Re: proof that {(1+x/n)^n} is bounded

Re: proof that {(1+x/n)^n} is bounded

hi dgomes

i think you have the wrong expression.i think his is : (1+(x/n))^n just written without the brackets.

Re: proof that {(1+x/n)^n} is bounded

Ok, you are right! At least this other example is done. I will think later on the correct one.

Did you find any mistakes on the example that I did?

Re: proof that {(1+x/n)^n} is bounded

i'm really tired,so even if there are mistakes i won't be able to catch them.but as far as i can see right now it's all pretty much correct.just to be sure you should ask someone else.