# Thread: Non Increasing and Decreasing Sequence

1. Let $\left\{{a}_{n}\right\}$ be a nonnegative, non-increasing sequence and convergence to $a \ge 0$. Can we say that ${a}_{n}\ge a$ for all n $\in \Bbb{N}$ ?

Also, if $\left\{{a}_{n}\right\}$ is a nonnegative, decreasing sequence and convergence to $a \ge 0$. Can we say that ${a}_{n}> a$ for all n $\in \Bbb{N}$ ?  Reply With Quote

2.

3. Originally Posted by ozkan12 Let $\left\{{a}_{n}\right\}$ be a nonnegative, non-increasing sequence and convergence to $a \ge 0$. Can we say that ${a}_{n}\ge a$ for all n $\in \Bbb{N}$ ?

Also, if $\left\{{a}_{n}\right\}$ is a nonnegative, decreasing sequence and convergence to $a \ge 0$. Can we say that ${a}_{n}> a$ for all n $\in \Bbb{N}$ ?
Hi ozkan12,

Let me give you a hint for the first part. Consider the sequence,

$\left\{a_n\right\}_{n=1}^{\infty}=\left\{1+\frac{(-1)^n}{n}\right\}_{n=1}^{\infty}$  Reply With Quote

İs this sequence convergent ?  Reply With Quote

5. Originally Posted by ozkan12 İs this sequence convergent ?
Take the limit of the sequence,

$\lim_{n\rightarrow\infty}\left(1+\frac{(-1)^n}{n}\right)=\lim_{n\rightarrow\infty}1+\lim_{n\rightarrow\infty}\frac{(-1)^n}{n}=1+\lim_{n\rightarrow\infty}\frac{(-1)^n}{n}$

What does that equal to?  Reply With Quote

limit of sequence equal to 1, but this sequence is not non-increasing...  Reply With Quote

7. Originally Posted by ozkan12 limit of sequence equal to 1, but this sequence is not non-increasing...
Hi ozakn12,

I guess this depends on what you mean by non-increasing. I took it as a sequence that is not increasing. For an each term should be greater than the one before it. Hence this sequence is non-increasing.

The first few terms of the sequence are, $\left\{0,\frac{3}{2},\frac{2}{3},\frac{5}{4},\frac{4}{5},\frac{7}{6},\frac{6}{7},\ldots\right\}$  Reply With Quote

Dear,

I dont understand, what you say...Can you explain my question ? Because your hint is very strange for me, İn most source, I see definition of increasing, non increasing, not increasing, decreasing, non decreasing vs......And there are very difference between these definitions....Therefore, I didnt understand these definitions...Can you explain my questions without any example...  Reply With Quote

9. Originally Posted by ozkan12 Dear,

I dont understand, what you say...Can you explain my question ? Because your hint is very strange for me, İn most source, I see definition of increasing, non increasing, not increasing, decreasing, non decreasing vs......And there are very difference between these definitions....Therefore, I didnt understand these definitions...Can you explain my questions without any example...
I guess first we need to get the definitions cleared out. It is true that the definitions can have minor differences according to what source you refer. Did you learn about sequences in school/university? If so could you please write down your definitions for increasing and decreasing sequences or point me to a specific source where you learnt these?  Reply With Quote

Dear,

I learnt these definitions from internet, some analysis book etc...I didnt decide that which is true ?

And which is true for non-increasing $\left\{{a}_{n}\right\}$ sequence which is convergent to nonnegative $p\ge 0$.

${a}_{n} \ge p$ for all $n\in \Bbb{N}$ or ${a}_{n} > p$ for all $n\in \Bbb{N}$. Which is true ?  Reply With Quote

11. Originally Posted by ozkan12 Dear,

I learnt these definitions from internet, some analysis book etc...I didnt decide that which is true ?

And which is true for non-increasing $\left\{{a}_{n}\right\}$ sequence which is convergent to nonnegative $p\ge 0$.

${a}_{n} \ge p$ for all $n\in \Bbb{N}$ or ${a}_{n} > p$ for all $n\in \Bbb{N}$. Which is true ?
Refer one specific source, so that you don't get mixed up with the definitions. A good source of reference to learn about the basic definitions of sequences is .  Reply With Quote

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