Measurable Spaces – Problem (43/365)

Continuing the last post on showing that certain sets are Borel sets, today we ask if the following is a Borel set.

$\displaystyle \{ x \in R^\infty : x_n \rightarrow \}$

This is the set of all sequences converging to a finite limit. My initial thought was to use the result from last time where we showed that the set of sequences bounded from above or below by $a$

$\displaystyle \{ x \in R^\infty : x_n \rightarrow a \} = \{ x \in R^\infty : \sup \inf x_n \ge a \} \cap \{ x \in R^\infty : \inf \sup x_n \le a \}$

But we can’t then union all the sets of the above form for each possible limit because there are uncountably many choices. It would seem that we need a way to characterize limits without picking the value of the limit. Luckily, there is such a characterization for converging sequences of real numbers; namely, Cauchy sequences. A sequence $x_1, x_2, \dots$ is a Cauchy sequence, if for all $\epsilon > 0$ , there is a positive integer $N$ such that for all $m,n > N$ , $| x_m - x_n | < \epsilon$ . All sequences of real numbers converging to a finite limit are also Cauchy sequences.

The Cauchy condition is true if and only if for all $m \ge 1$ , $\lim_n | x_n - x_{n+m} | = 0$ . We can write the set of all converging sequences as

$\displaystyle \cap_{m=1}^\infty \{ x \in R^\infty : | x_n - x_{n+m} | \rightarrow 0 \} \\ \cap_{m=1}^\infty \{ x \in R^\infty : \sup_n \inf_{k \ge n} | x_k - x_{k+m} | \ge 0 \} \cap \{ x \in R^\infty : \inf_n \sup_{k \ge n} | x_k - x_{k+m} | \le 0 \}$

As a result, the set of all sequences converging to a finite limit is measurable.

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