Reverse a Stack Using Recursion

I am back! I will talk about how to reverse a stack using recursion (note: recursion means we need the function call stack....)
Given a stack S=<0, 1, 2, ..., 8, 9>, where 9 is the topmost element. The reversed stack S' is <9, 8, ..., 2, 1, 0>.  The basic idea is to reverse the smaller stack subS=<0, 1, 2, ..., 8> first to get subS'=<8, ..., 2, 1, 0>, and then put 9 at the bottom of subS' to get S'. So:

void reverse_stack(stack<int> &s){
    if(!s.empty()){
        int ele = s.top();
        s.pop();
        reverse_stack(s);
        put_bottom(s, ele);
    }
}

But how to put an element at the bottom of a stack? we still can use recursion!

void put_bottom(stack<int> &s, int ele){
    if(s.empty()){
        s.push(ele);
        return;
    }
    int topEle=s.top();
    s.pop();
    put_bottom(s, ele);
    s.push(topEle);
}