How to increase number of push ups a day?
Keep doing push-ups daily! Do your maximum number of reps till failure, rest for a minute, and then repeat at least two more sets. Repeat this daily and you should see marked improvements within just a week.
If you're feeling like you shouldn't push yourself too hard, make sure you complete at least two sets a day, till failure, and rest for a day every two days or so.
O good way is to do some top-down-top workout.
One day do all you can, (lets say 20).
On the next day, do this:
(start with half pushups of your maximum, in this example 10)
10 Push, rest 5 to 10 secs
8 push, Rest 5 to 10 secs
7 push, rest 5 to 10 secs
And so on so you get to 1
An then start incrementing
1 push, rest 5 to 10 secs
2 push, Rest 5 to 10 secs
So on, until you get again to 10 push.
Do this for one week, and the next week, do 2 rounds. In the 3rd week start all over again (do as much as you can, the next day start the workout with half of your maximum).
This will help you to gain strength, and in short time you'll be doing a lot pushups in a row.