Science cannot prove anything, it only obtains evidence to support theories. Math proves things, given certain postulates.
That said, what do you mean by the existence of sentimental value? That people attach values to objects that are not inherent in their physical properties? That a child will treat a stuffed rabbit as a friend, and when seeing the toy ten years later they will break down crying because it embodies their childhood and happiness lost? One could observe this, and say that it happens. Through science one could conduct an experiment and see how, in certain situations, people tend to assign emotional worth to objects, and say that this supports the existence of sentimental value.
But I’m not quite sure what you’re asking.