Since k is an integer, then the sum of 2k-1 and 2k is an integer.  Getting its sum, we have 2k-1 + 2k = 4k - 1

4k - 1 is an integer.
4k is an even integer, becase k is an integer, and it is a multiple of 2: 4k = 2(2k). 
being a multiple of 2 is the condition for an integer to become even.

Thus, 4k-1 is an odd integer because subtracting 1 from an even integr makes it an odd one because it will no longer be a multiple of 2