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.

Direct Proof Example : If n is even, then n + 1 is odd

This will be the first example we will encounter as this expedition for finding out the truth of many mathematical and logical mysteries on our journey into the realm of unsolvable problems.
Our first problem is:
Prove: If n is even, then n + 1 is odd

Proof: (Direct)
Assume n is even, therefore n has the form of an even integer which is 2k, and k is any integer.

Calculate:
n = 2k
add one to both sides
n + 1 = 2k + 1
by definition 2k + 1 is an odd integer and it equals n + 1

Conclusion:
This direct proof has shown that if n is even then n + 1 must be odd.

I have posted the indirect proof that rounds off this simple proof that wets our appetite for knowledge.