<s>[INST] Demonstrate that the sequence is decreasing $$\left( 1-\frac{1}{n+1}\right)^n$$
How do I demonstrate that the sequence decreases? I tried using the newton binomial but I end up with a terribly complicated expression. 
 [/INST] This is a classic exercise.
Let 
$$a_n= \left( 1-\frac{1}{n+1}\right)^n=\left( \frac{n}{n+1}\right)^n$$
$$\frac{a_n}{a_{n+1}} =\left( \frac{n}{n+1}\right)^n\left( \frac{n+2}{n+1}\right)^{n+1}=\left( \frac{n}{n+1}\right)^{n+1}\left( \frac{n+2}{n+1}\right)^{n+1}\frac{n+1}{n}$$
Now write 
$$\left( \frac{n}{n+1}\right)^{n+1}\left( \frac{n+2}{n+1}\right)^{n+1}=\left( \frac{n^2+n}{(n+1)^2}\right)^{n+1}=\left( 1-\frac{1}{(n+1)^2}\right)^{n+1}$$
and use Bernoulli inequality. 
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