<s>[INST] Proving of Boundedness and Monotonicity of a sequence $e(n) = (1 + \frac 1n)^n$ is convergent. How do we prove it?(proofs other than binomial expansion are welcome)
 [/INST] By Bernoulli's inequality, we have $$\left(1-\frac 1{(n+1)^2}\right)^{n+1}\ge 1-\frac 1{n+1}=\frac n{n+1}$$ Now note that $$1-\frac 1{(n+1)^2}={n^2+2n\over n^2+2n+1}={n(n+2)\over (n+1)^2}=\frac {n+2}{n+1}\frac n{n+1},$$ so the above inequality reads $$\left(\frac {n+2}{n+1}\right)^{n+1}\left(\frac n{n+1}\right)^{n+1}\ge \frac n{n+1}$$ Multiplying both sides with $\displaystyle\left(\frac {n+1}n\right)^{n+1}$ gives you the monotonicity of the sequence.
To prove boundedness, use the binomial theorem to show $\left(1+\frac 1n\right)^n\le 3$ noting that $${n\choose k}\left(\frac 1n\right)^k=\frac 1{k!}{n(n-1)...(n-k+1)\over n^k}=\frac 1{k!} \frac{n}n\frac{n-1}n...\frac{n-k+1}n\le \frac 1{k!}\le \left(\frac 12\right)^{k-1}.$$
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