Bounded sequences and convergent subsequences in metric spaces

[AxiomOfChoice]

Suppose we're in a general normed space, and we're considering a sequence $\{x_n\}$ which is bounded in norm: $\|x_n\| \leq M$ for some $M > 0$. Do we know that $\{x_n\}$ has a convergent subsequence? Why or why not?

I know this is true in $\mathbb R^n$, but is it true in an arbitrary normed space? In particular, since it's true in $\mathbb R$, we know that $\{\|x_n\|\}$ has a convergent subsequence $\{\|x_{n_k}\|\}$ that converges to some $z\in \mathbb R$. My first instinct would be to try to apply the triangle inequality to show that $\|x_{n_k} - x\| \to 0$ for $\|x\| = z$, but the triangle inequality doesn't give me what I want here, since I need to bound $\|x_{n_k} - x\|$ by something that can be made arbitrarily small, but I only have $| \|x_{n_k}\| - \|x\| | \leq \|x_{n_k} - x\|$.

 

[disregardthat]

Take the space Q^n over Q, and you will see that it is not the case. E.g. in Q, take an increasing sequence converging to sqrt(2) in R. Any subsequence will converge to sqrt(2) in R, so it will not converge to any element of Q.

 

[mathman]

Suppose we're in a general normed space, and we're considering a sequence $\{x_n\}$ which is bounded in norm: $\|x_n\| \leq M$ for some $M > 0$. Do we know that $\{x_n\}$ has a convergent subsequence? Why or why not?

I know this is true in $\mathbb R^n$, but is it true in an arbitrary normed space? In particular, since it's true in $\mathbb R$, we know that $\{\|x_n\|\}$ has a convergent subsequence $\{\|x_{n_k}\|\}$ that converges to some $z\in \mathbb R$. My first instinct would be to try to apply the triangle inequality to show that $\|x_{n_k} - x\| \to 0$ for $\|x\| = z$, but the triangle inequality doesn't give me what I want here, since I need to bound $\|x_{n_k} - x\|$ by something that can be made arbitrarily small, but I only have $| \|x_{n_k}\| - \|x\| | \leq \|x_{n_k} - x\|$.

The answer is trivially no. Example in Hilbert Space:
Let x_n=(0,0,...,1,0,0...) where the 1 is the nth coordinate. Then ||x_n||=1, but the sequence has no convergent subsequences.

The underlying difference is that bounded sets in finite dimensional spaces have a compact closure, but not necessarily in infinite dimensional spaces.

 

[HallsofIvy]

Yet another example: $a_1= 3$, $a_2= 3.1$, $a_3= 3.14$, etc. where each term is one more decimal place in the decimal expansion of $\pi$. That is a bounded sequence of rational numbers. Thinking of it as a sequence in the real numbers, it converges to $\pi$ which means every subsequence converges to $\pi$. Thinking of it as a sequence in the rational numbers, clearly no subsequence can converge to a rational number.

More generally, a sequence of rational numbers that converges to an irrational number, cannot have any subsequence that converges to a rational number. Thus, thinking of the sequence as in the rational numbers, no subsequence can converge.

 

[disregardthat]

Halls that is more or less the same example as what I brought up.

 

[AxiomOfChoice]

Thanks a lot, guys. I should have thought of the "orthonormal sequence in an infinite dimensional Hilbert space" example. Oh well...now I know (hopefully)!