- #1
bw0young0math
- 27
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hello. This is my first post.
Of course I read rules, but I may make mistakes about posting.
If I have a mistake about something, please tell me.
Now, this problem15 is in section 5.1 from Bartle.
f: (0,1)→R be bounded but such that x→0,lim f does not exist.
Show that there are two sequences (x_n), (y_n) in (0,1) with lim(x_n)=0=lim(y_n),
but such that lim f(x_n) and lim f(y_n) exist but are not equal.
I tried to understand that problem. so I tried to take an example at first.
The example is here. f(x)=sin(1/x)
If I take two sequences,x_n=1/2nπ and y_n=1/(2nπ+π/2),
lim f(x_n)=0, lim f(y_n)=1. Thus, they are not equal.
hum...
Second, I tried to solve that problem by using logic. However, I coudn't.
That problem is so complex that I couldn't change that problem to logic easily.
I want to use reduction to absurdity.
Is negation of "there are two sequences (x_n), (y_n) in (0,1) with lim(x_n)=0=lim(y_n),
but such that lim f(x_n) and lim f(y_n) exist but are not equal."
"Every sequence (x_n) with lim(x_n)=0 satisfies lim f(x_n)=L exist and they are equal." right?![Confused :confused: :confused:](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
Of course I read rules, but I may make mistakes about posting.
If I have a mistake about something, please tell me.
Now, this problem15 is in section 5.1 from Bartle.
f: (0,1)→R be bounded but such that x→0,lim f does not exist.
Show that there are two sequences (x_n), (y_n) in (0,1) with lim(x_n)=0=lim(y_n),
but such that lim f(x_n) and lim f(y_n) exist but are not equal.
I tried to understand that problem. so I tried to take an example at first.
The example is here. f(x)=sin(1/x)
If I take two sequences,x_n=1/2nπ and y_n=1/(2nπ+π/2),
lim f(x_n)=0, lim f(y_n)=1. Thus, they are not equal.
hum...
Second, I tried to solve that problem by using logic. However, I coudn't.
That problem is so complex that I couldn't change that problem to logic easily.
I want to use reduction to absurdity.
Is negation of "there are two sequences (x_n), (y_n) in (0,1) with lim(x_n)=0=lim(y_n),
but such that lim f(x_n) and lim f(y_n) exist but are not equal."
"Every sequence (x_n) with lim(x_n)=0 satisfies lim f(x_n)=L exist and they are equal." right?