- Thread starter
- #1

#### bw0young0math

##### New member

- Jun 14, 2013

- 27

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?????????