INDIRECT PROOF: If n is even, then n + 1 is odd

Even though the statement was previously shown true using Direct Proof, we should still prove our statement even further. Another method is indirectly proving a statement. To accomplish this task we take the statement, "If n is even, then n + 1 is odd," and use the contrapositive and prove "If n is odd, then n + 1 is even."
Here is the formal proof :

Prove: If n is even, then n + 1 is odd
Proof : (Indirect)
//our mental note of the contrapositive:  "If n is odd, then n + 1 is even"
Assume n is odd, therefore n has the form of an odd integer which is 2k - 1, where k represents any integer.

Calculate:
n = 2k - 1
add one to both sides
n + 1 = 2k
by definition 2k is even
therefore n + 1 = (an even number)

Conclusion:
We have shown indirectly that the contrapositive is true therefore the original must be true

We have successfully proved the statement "If n is even, then n + 1 is odd," both Directly and Indirectly. We will continue with more basic proofs while expanding on each one. If any questions should arise, comment and I will reply as soon as possible.