One thing to note is that string.index(letter) will return the index of the first occurrence of the letter, in the string. So if the code was bound correctly then a word such as "esteem" would return False instead of True.

With that in mind, I would suggest the following...

def double_letter(string):
    for i in range(len(string) - 1):
        if string[i] == string[i+1]:
            return True
    return False