sincerely
In a real, honest way, with true feelings and meaning.
Sincerely means genuinely, honestly, or from the heart. When you sincerely apologize to a friend you've hurt, you truly mean it. You actually feel sorry and want to make things right, rather than saying words to get out of trouble or because someone told you to.
You can sincerely believe something, like when you're genuinely convinced your soccer team will win the championship. You can sincerely hope for something, like truly wishing your sick grandmother will feel better. The key is that sincerity involves real feelings with no pretending, no acting, no hidden motives.
People can usually tell the difference between sincere feelings and fake ones. A sincere compliment makes you feel good because you know the person means it. An insincere compliment feels hollow and awkward. When you thank someone sincerely, they can hear the real gratitude in your voice.
Sincerely also appears at the end of formal letters, right before your signature. It signals that everything you wrote came from honest intentions.