Exercise 1.14

Show that in the standard metric on the line, $$\frac{1}{n} \rightarrow 0$$ and that the sequence $$x_n = (-1)^n$$ does not converge to any point.

Masteranza
(No better proposal for the first part)

Prove that $$x_n = (-1)^n$$ does not converge to any point
Let's notice that $$d(x_{n+1},x_n) = 2$$ for all $$n $$, so any ball that we could construct would need to have it's radius $$\epsilon \geq 1$$

Masteranza 20:51, March 6, 2011 (UTC)

''' Slick. Since the ball around L would need to contain all $$x_n$$ for $$n$$ sufficiently large, it will need to contain $$x_N$$ and $$x_{N+1}$$ for some N, and thus its radius must be strictly greater than 2, in fact. --Steven.clontz 21:53, March 9, 2011 (UTC)'''

Prove that 1/n -> 0
$$\frac{1}{n} \rightarrow 0$$ is equivalent to the following statement: $$\forall \epsilon > 0 \ \exists N \in \mathbb{N} : n \geq N \rightarrow \frac{1}{n} \in B(0,\epsilon)$$. Using the standard metric, $$\frac{1}{n} \in B(0,\epsilon)$$ is equivalent to $$\|\frac{1}{n}\| < \epsilon$$. Constructing an N for any $$\epsilon$$ such that $$n \geq N$$ implies the prior statement will complete the proof. $$\frac{1}{n} < \epsilon \rightarrow \frac{1}{\epsilon} < n$$, therefore $$N = \lceil \frac{1}{\epsilon} \rceil$$ satisfies the requirements for $$\frac{1}{n}$$ to converge to 0.

'''Looks good! --Steven.clontz 21:54, March 9, 2011 (UTC)'''

Prove that $$x_n = (-1)^n$$ does not converge to any point
Using the standard metric, $$d(-1,1) = 2$$. Assume that $$x_n \rightarrow L$$. Given some $$\epsilon < 1$$, assume that for some $$N$$ that $$d(x_N,L) < \epsilon$$. Since $$x_n$$ alternates between -1 and 1, $$d(x_{N+1},L) = 2 - d(x_N,L)$$ for $$L \in [-1,1]$$ or $$d(x_{N+1},L) = 2 + d(x_N,L)$$ for $$L \not\in [-1,1]$$. Either way, since $$d(x_{N+1},L)$$ is guaranteed to be greater than 1, it is immpossible to find an $$N \in \mathbb{N}$$ such that $$n \geq N \rightarrow x_n \in B(L,\epsilon)$$ for any $$L$$, therefore $$x_n$$ does not converge to any point.

'''This works. --Steven.clontz 21:56, March 9, 2011 (UTC)'''