Journal 4 - Week 5

Friday   October. 10th   Sunny

This week is special, since we had our term test 1. Questions appeared in test 1 are so great that they are obviously a proper review for what we have learned until today. Thus, in this journal, I will talk about both contents covered in our lectures and in term test 1.

Firstly, as usual, I will introduce the materials which are new to us from this week. We learned several typical examples for proofs, such as proof using contrapositive, proof using contradiction, proof for existence, and proof about a sequence.

Secondly, I will talk about the hardest part this week and how I am trying to make it easier for myself. Sequence is not just a regular number, it is a bunch of numbers which are related in some way. This means that when we are proving something about a sequence, we can not just prove only one number and we have to prove all the numbers in the sequence set. This new concept challenged me, so I did a lot of exercises about sequence proof and feel much better for now.

Thirdly, I chose one of the proof questions in my exercise, which follows as below.

Q: Sequence: ∀n∈N, a_n = n²

Now prove: ∃i∈N, ∀j∈N, a_j ≤ i ⇒ j < i

A graph could make the sequence look more clear. (point when n = 0, 1, 2, 3, 4, …)

As the graph shows, we can choose i when i is equal to 2 to help us in our proof. 

Proof:

Let i = 2. Then i∈N.   # 2∈N

Assume j∈N.

       Assume a_j ≤ i.   # antecequent

              Then j² ≤ 2   # by definition

              Then j ≤ √2

              Then j ≤ √2 < 2

       Then a_j ≤ i ⇒ j < i

Then ∀j∈N, a_j ≤ i ⇒ j < i

Then ∃i∈N, ∀j∈N, a_j ≤ i ⇒ j < i

Lastly, our tutorial of this week focused on the detailed proof structures for statements. Although we are not required to write complete proofs for our two exercise questions, these questions still help us to improve our ability to deal with our proofs. If we can clearly know the structure of each proof, we would not lose marks on basic rules of proof. Also, we would have more time on thinking about the question itself.

Much appreciated for reading my journal.
See you soon :)