Two methods to do it:

Method 1:

There are 99999 positive integers from 1 through 99999.
But 9999 of them have 4 or fewer digits.

Therefore there are 99999-9999 = 90000 5-digit integers.

Method 2:

Choose the first digit any of 9 ways, from 1 through 9 inclusive.
Choose the second digit any of 10 ways, from 0 through 9 inclusive.
Choose the third digit any of 10 ways, from 0 through 9 inclusive.
Choose the fourth digit any of 10 ways, from 0 through 9 inclusive.
Choose the fifth digit any of 10 ways, from 0 through 9 inclusive.

Answer: 9*10*10*10*10 = 90000 5-digit positive integers.

Edwin