janw@inmet.UUCP (10/08/85)
A continuous function is uniquely defined by its values over rational arguments. Rational numbers are countable; thus a continuous real function is uniquely defined by a sequence of real numbers. To answer the problem, it is enough to assign every sequence of real numbers a unique real number. We can assume all the numbers to be in [0,1). Then a sequence S of them can be represented by an infinite matrix of digits such that s[i,j] is the digit in the i-th position in the j-th member of the sequence. Traversing this matrix in the usual diagonal fashion : concatenating the finite diagonals in the order of increasing length (s[1,1]s[2,1]s[1,2]s[3,1] ...), we obtain a digit sequence uniquely defined for the matrix; pre- ceded by "0.", this yields the real number corresponding to S. Jan Wasilewsky
janw@inmet.UUCP (10/09/85)
> >> Since the cardinality of the set of polynomials > >> mapping R -> R is C, and any continuous function > >> mapping R -> R is the limit of a sequence of >**>> polynomials (the card of the set of such sequences >**>> again being C), therefore the card of the set of > >> continuous functions is also C. >Nope. Consider: Each real number is the limit of a sequence of >rational numbers, but reals are uncountable and rationals are. >[etc.] The marked proposition in parentheses is perfectly correct. To prove this, it is sufficient to assign a unique real number to every sequence of real numbers. (And then apply this twice: to the sequence of coefficients of a polynomial, then to the sequence of polynomials). Proof follows: We can assume all the numbers to be in [0,1) (the extra step of going from this to unlimited numbers is quite easy). Then a sequence S of them can be represented by an infinite matrix of decimal digits such that s[i,j] is the digit in the i-th position in the j-th member of the sequence. Traversing this matrix in the usual diagonal fashion : concatenating the finite diagonals in the order of increasing length (s[1,1]s[2,1]s[1,2]s[3,1] ...), we obtain a digit sequence uniquely defined for the matrix; pre- ceded by "0.", this yields the real number corresponding to S. Jan Wasilewsky