$*37.$ Prove that R$2$ R$.[$Hint: Think of shuffling two digit strings

$($a$1$a$2$a$3...)$&$($b$1$b$2$b$3...)->($a$1$b$1$a$2$b$2$a$3$b$3...).$

In this way you could transform a pair of reals into a single real. Be sure to

face the nines$-$termination issue.$]$