I have the urge to tell my supervisor that I can do it.
Why do you feel the need to tell your supervisor that?
I just wanted to secure my job.
It's great to try to secure your job then by doing what you need to do. your supervisor would be proud of you, but you need to not be a kiss up by constantly overdoing it. There's a limit to everything.
I'm going to say hi to my supervisor every time I see her.
It's nice to be friendly, don't be too overbearing. I recommend keeping up the good work and letting them know that you are up to any task that is needed to be done.