- #1

cbarker1

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MHB

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In my book, Basic Analysis by Jiri Lebel, the exercise states

"show that the sequence $\left\{(n+1)/n\right\}$ is monotone, bounded, and use the monotone convergence theorem to find the limit"

My Work:

The Proof:

Bound

The sequence is bounded by 0.

$\left|{(n+1)/n}\right| \ge 0$

$n+1\ge0\, \forall\, n \in \Bbb{N}$

Monotone

$\frac{n+1}{n} \ge 0$$\frac{\left(n+1\right)+1}{n+1}\le\frac{n+1}{n}$

$\frac{n+2}{n+1}\le\frac{n+1}{n}$

$n^2+2n\le n^2+2n+1$

the sequence is increasing monotone.

Applying the theorem.

$\lim_{{n}\to{\infty}} \frac{n+1}{n}=\sup{{\frac{n+1}{n}:n\in\Bbb{N}}}$

Let A=$\sup{{\frac{n+1}{n}:n\in\Bbb{N}}}$

We know the sup A is less than or equal to 1 as 1 is an upper bound. Take a number $$b\le1$$ such that $$b\ge \frac{n+1}{n}$$.

$$bn\ge n+1$$

$$bn-n\ge1$$

$$n(b-1)\ge1$$

By the Archimedean Property..."

What need I to do in order to finish the proof?

THanks

CBarker1