Excel - Check if numerical day is between two dates

Good afternoon SO community,

Here is a little setup to help everybody understand the situation.

Sheet1!B3 has the date 'Tuesday, July 26, 2016'
Sheet1!B4 has the date "Tuesday, August 02, 2016'

Sheet2!C5 has the value of 1.

The value in Sheet2!C5 represents the day of the month. For example, this case would be the first of the month. I'm trying to figure out how to check to see if the value in Sheet2!C5 is between the dates in Sheet1!B3 and Sheet1!B4.

Here's what I've tried:

=IF(AND(Sheet2!C5>DAY(B3),Sheet2!C5<(DAY(B3)+7)),"True","False")

=IF(AND(Sheet2!C5>B3,Sheet2!C5<(B3)+7),"True","False")

=IF(AND(Sheet2!C5>DAY(B3),Sheet2!C5<DAY(B4)),"True","False")

=IF(AND(Sheet2!C5>B3,Sheet2!C5<B4),"True","False")

None of those formulas work due to the fact that the day of the month would end at 31 or 30 and restart at 1.

Does anybody have any ideas on what I could try to get this to work?

Thank you in advance, Tim