How to wrap tight a cut finger a child is picking on?

I cut my finger and have some questions.?

• I sliced my finger yesterday with a pair of scissors trying to cut the top off a bottle, it happened when i was trying to make the initial incision and it just pierced the other side and the hand that was holding it. Anyways it was bleeding pretty bad and it ran to the washroom and bled into the sink, it just kept on bleeding and i ran some water over it and it continued to gush out, then i pinched the cut with my other hands for 5 minutes and the bleeded subsided. Then I dried my hand off and put polysporin on it and wrapped it in cause and left it on for the rest of the day and night. I opened it just now and reapplied some polysporin and gonna let the cut be exposed to the air. I also cut my finger on an angle so its not totally and open wound, the deepest part of the cut which is underneath skin appears to be a dark red/black line which i assume is dried up blood. I should add that all i did to clean it was run water over it, i didn't have any rubbing alcohol or anything. So now I'm thinking I should go to the pharmacy and buy some, should I? its slightly swollen on my finger tips (cut is on the side of my finger) Thanks for any help.

• Answer:

  I prefer peroxide as it doesn't sting but doctors always recommend neosporin. Either can be found at the pharmacy and of course keep a bandaid on it until healed.