We're demonstrating that we've solved the problem as stated, not demonstrating that the halting problem can be solved. There is no time frame for how long your test program can run (in the clients wishes), so even if there is an infinite loop in the program fed in - your own test program also waits infinitely. Because your program never stops, and is basically still doing it's job, the client cannot tell you that your solution is wrong. He can only tell you that your solution is wrong if you return a wrong answer, but you never do.

So while the halting problem cannot be solved, the clients requirements can be fulfilled (logically).