Surjective Function from a Cantor Set
I recently solved an interesting problem on an midterm. Here is one piece of it:
Let $K =\prod_{i=1}^\infty \{0,1\}$ in the product topology, and let $s_1,s_2,\ldots$ be a sequence of positive real numbers such that $\sum_{i=1}^\infty s_i = 1$. Define a function $f\colon K \to [0,1]$ by $(k_1,k_2,\ldots) \mapsto \sum_{i=1}^\infty k_is_i$.
Show that $f$ is continuous. (This part I had no trouble with.)
Show that if $s_k \leq \sum_{i=k+1}^\infty s_i$ for all $k$, then $f$ is surjective.
I was able to prove this, but I'm not satisfied with my proof of the second part, because it seems like there should be a much easier way to show this. Does anyone see a simple/simpler approach to this problem than mine?
Here's a very rough outline of my proof. I've left out a lot of details, so let me know if something really doesn't make sense.
Suppose there is some $x \in [0,1] \setminus f(K)$. Then there is a neighborhood $(x-\epsilon,x+\epsilon)$ which isn't hit by $f$. Since $K$ is compact, $f(K) \cap [0,x]$ is compact, hence closed, and so it contains its supremum. This supremum is the image of an element of the form $(k_1,\ldots,k_n,0,1,1,\ldots)$. By our hypotheses we get $f((k_1,\ldots,k_n,0,0,\ldots)) < x-\epsilon$ and $f((k_1,\ldots,k_n,0,1,1,\ldots)) > x+\epsilon$, so we take the supremum of the images of all elements that start with $(k_1,\ldots,k_n,0)$ and map into $[0,x]$. This element is again the image of an element whose entries are eventually equal to 1. Repeating this gives a sequence of pairs of elements of $K$ with longer and longer initial parts. These must converge in $K$, but their images have to be at least $2\epsilon$ apart, so we reach a contradiction.
Solution 1:
It's simpler to show explicitly, for an arbitrary $x\in[0,1]$, how to construct $(k_n)_n\in K$ that maps to $x$.
Define $k_n$ recursively by $k_i=1$ iff $(\sum_{j=1}^{i-1} k_j s_j)+s_j \le x$ and $k_i=0$ otherwise ...
Solution 2:
The question already has been answered. I just wanted to mention, that this result (or a very similar one) appears in mathematical literature.
Ribenboim P. My Numbers, My Friends, Springer, 2000, p. 51.
Let $(s_i)$ be a sequence of positive real numbers such that $s_1>s_2>s_3>\dots$ and $\lim\limits_{i\to\infty} s_i=0$. Let $S=\sum_{i=1}^\infty s_i=S\le+\infty$. We say that $x > 0$ is representable by the sequence $(s_i)$ if $x=\sum_{j=1}^\infty s_{i_j}$ (with $i_1<i_2<i_3<\dots$ ). Then, necessarily $x\le S$.
The first result is due to Kakeya; for the sake of completeness, we give a proof:
Proposition 1. The following conditions are equivalent:
- Every $x$, $0 < x < S$, is representable by the sequence $(s_i)$, where $i_1$ is the smallest index such that $s_{i_1}<x$.
- Every $x$, $0 < x < S$, is representable by the sequence $(s_i)$.
- For every $n\ge 1$, $s_n\le \sum_{i=n+1}^\infty s_i$.
See also:
- J. A. Fridy. Generalized bases for the real numbers. Fibonacci Q., 4:193–201.
- J. L. Brown. On generalized bases for real numbers. Fibonacci Q., 9:477–496.
- C. R. Banerjee, B. K. Lahiri: On subseries of divergent series; American Mathematical Monthly, 1964
Proof of the above result is basically the greedy approach. If we already have constructed some partial sum $\sum_{k=1}^{n-1} s_i\le x$, $i_n$ will be the smallest index such that $s_i\le x-\sum_{k=1}^{n-1} s_k$. The proof can be found also here. Note that the proof works for non-increasing sequences. (Although the formulation in Ribenboim's book uses decreasing sequences only.)
As can be seen from Henning's answer, the condition that $(s_k)$ is non-increasing is not really needed for this claim to hold.
Alternatively, we can reorder the sequence $s_k$ starting from the largest element. Such reordering cannot influence the validity of condition $s_n\le \sum_{i=n+1}^\infty s_i$. Since $\lim\limits_{i\to\infty} s_i=0$, such reordering is possible. (There are only finitely many terms greater than 1/2, finitely many terms greater than 1/4, etc.)