
#1
September 10th, 2015,
13:42
Can we say nonincreasing equivalent to decreasing (For sequence or functions) ? İn some book, I see that...Can you say something ?

September 10th, 2015 13:42
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#2
September 10th, 2015,
13:58
Nonincreasing includes the category of constant, whereas I would think many authors would not allow that for a decreasing function.
It's much like the following:
\begin{align*}
\text{Nonnegative integers}&=\{0,1,2,\dots\} \\
\text{Positive integers}&=\{1,2,3,\dots\}
\end{align*}

#3
September 10th, 2015,
14:02
Thread Author
I didnt understand dear Ackbach...What I wonder is nonincreasing equivalent to decreasing ?

#4
September 10th, 2015,
14:11
They're not equivalent in my mind. A horizontal line like $y=f(x)=2$ is nonincreasing. That is, it's not getting bigger. But it's also not decreasing  getting smaller. So a horizontal line would be in the set of nonincreasing functions, but it would not be in the set of decreasing functions. So the set of nonincreasing functions is not equal to the set of decreasing functions. You could define these two sets this way:
\begin{align*}
\text{nonincreasing functions}&=\{f(x_1<x_2) \implies (f(x_2)\le f(x_1))\} \\
\text{decreasing functions}&=\{f(x_1<x_2)\implies (f(x_2)< f(x_1)) \}.
\end{align*}
One has strict inequality  the decreasing functions  and one does not.

#5
September 10th, 2015,
14:27
Thread Author
İn some sources, I see that if f function is non increasing, then f is decreasing.
İf knowledges in this souce is true, can we say same thing for sequence ?

#6
September 10th, 2015,
15:03
Originally Posted by
ozkan12
İn some sources, I see that if f function is non increasing, then f is decreasing.
İf knowledges in this souce is true, can we say same thing for sequence ?
This is strictly by author. You have to know how the author you're reading defines these terms. It's the true the Wiki seems to define nonincreasing and decreasing as the same thing. I would find that confusing. I know other authors do not define it the same way. I'm afraid I'm not able to say much beyond this, other than an author would probably use the same sort of definition for sequences as for functions.

#7
September 10th, 2015,
15:08
Thread Author
This is very confusing for me...I search these definitions but there is many definitions but they have different properties...I dont know what I do ??? Thank you for your attention dear...

#8
September 10th, 2015,
15:34
Originally Posted by
ozkan12
This is very confusing for me...I search these definitions but there is many definitions but they have different properties...I dont know what I do ??? Thank you for your attention dear...
You have to interpret in context. You work in someone's book like Kirkwood or Rudin, you find their definition, and you go with that while you're working in that book. What book are you using? And how does that author define increasing or decreasing functions? And how does the word "monotone" fit in?

#9
September 10th, 2015,
15:42
Thread Author
I work on some analysis books and on İnternet resources...But as I say, I found different things..

#10
September 10th, 2015,
16:07
Originally Posted by
ozkan12
I work on some analysis books and on İnternet resources...But as I say, I found different things..
This isn't a problem. When you're reading one author, always use his definition. When reading another author, use his definition. Don't mix and match, and you'll be fine.