Continuing the last post on showing that certain sets are Borel sets, today we ask if the following is a Borel set.
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
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 is a Cauchy sequence, if for all , there is a positive integer such that for all , . 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 , . We can write the set of all converging sequences as
As a result, the set of all sequences converging to a finite limit is measurable.