All that we have proven so far is that limit $(1+1/n)^n$ exists and considered to be a number 'e' which belongs to $(2,3)$
We haven't proven that 'e' is irrational or that lim $(1+(x/n))^n) = e^x$
We only have the properties of sequences like Monotone convergence theorem and basic properties to prove this.
I was able to prove the previous question
$((1+(1/n))^{2n})$ by using the theorem of sequences that says square of a convergent sequence converges to the square of the original limit.
But I believe that the convergence of this requires us to use
$(1+(x/n))^n) = e^x$
So, if some one can prove it by basic properties of sequences, please do so!
Best Answer
$$\left(1-\frac1n\right)^n = \left(\frac{n-1}n\right)^n = \frac 1{\left(\frac{n}{n-1}\right)^n} = \frac 1{\left(\frac{n-1+1}{n-1}\right)^n} = \frac 1{\left(1+\frac1{n-1}\right)^n} = \frac 1{\left(1+\frac1{n-1}\right)^{(n-1)n/(n-1)}} = \cdots $$