How can I create an arithmetic loop in a POSIX shell script?

I know how to create an arithmetic for loop in bash. How can one do an equivalent loop in a POSIX shell script? As there are various ways of achieving the same goal, feel free to add your own answer and elaborate a little on how it works. An example of one such bash loop follows: #!/bin/bash for (( i=1; i != 10; i++ )) do echo "$i" done