Why is $2-(1/(n+1)(1-(1/(n+1)) \leq 2-(1/(n+1)) $?

Question

In a proof by mathematical induction, the next step is given:

Why is $$2-\dfrac{1}{n+1}\left(1-\dfrac{1}{n+1}\right) \leq 2-\dfrac{1}{n+1} $$?

I will also upload the whole solution:

Answer 1

As TheSimpliFire has pointed out, the book made a mistake.

Instead, you just need to prove $2-{1\over n}+{1\over(n+1)^2}\le2-{1\over n+1}$. This is equivalent to proving

$${1\over(n+1)^2}\le{1\over n}-{1\over n+1}={1\over n(n+1)}$$

which is clear, since ${1\over n+1}\le{1\over n}$.

Answer 2

Your book is wrong. We have $$2-\frac1{n+1}\left(1-\frac1{n+1}\right)=2-\frac1{n+1}+\left(\frac1{n+1}\right)^2>2-\frac1{n+1}$$ for all $n\in\mathbb{N}$.