It's not quite true that you cannot use "any" in a positive sentence. It's usually taught that way, because that is the usual rule we follow, but actually the choice between "some" and "any" is subject to much more subtle rules.
He welcomed any stray cats into his home.
He welcomed some stray cats into his home.
The first sentence means that, as a matter of principle, if a stray cat came to his home, he would not turn it away. The second sentence means that at some definite time in the past, he welcomed an unspecified number of stray cats into his home, but this doesn't mean he would always do this.
Or compare these two sentences:
Do you have some beer?
Do you have any beer?
Spoken in a neutral tone, both these sentences mean roughly the same thing. But if I put special emphasis on the second sentence, I can make it express incredulity or frustration: "I can't believe you have no beer! What kind of a host are you?"