I am experiencing some conceptual difficulties in part (a) exercise 2.4.7 of Understanding Analysis by Stephen Abbott (2015), and would appreciate some clarity concerning proof by induction.
Context.
Exercise 2.4.7 (Limit superior).
Let $(a_n)$ be a bounded sequennce.(a) Prove that the sequence defined by $y_n = \sup \{a_k : k \geq n \}$ converges.
In order to prove this using tools developed in that section of the book, I opted for using the monotone convergence theorem, which requires showing that $(y_n)$ is bounded and decreasing. Where I'm experiencing doubt is in showing that $(y_n)$ is decreasing using induction.
Proof by induction that $(y_n)$ is decreasing.
Base case $y_1 \geq y_2$. Because $y_1 = \sup \{a_k : k \geq 1 \}$, it is also an upper bound for $\{a_k : k \geq 2 \}$. Because $y_2 = \sup \{a_k : k \geq 2 \}$ is the least upper bound for $\{a_k : k \geq 2 \}$, it follows that $y_1 \geq y_2$.
Inductive step. Assume $y_{k} \geq y_{k+1}$ for some $k \in \mathbb{N}$, and call this statement $S(k)$.
As I understand, I need to deduce the truth of $S(k+1)$ from the assumption that $S(k)$ is true. But this is where I am experiencing doubt. To me, it seems that I can show the truth of $S(k+1)$, that $y_{k+1} \geq y_{k+2}$, using the same argument as in the base case and without needing to assume that $S(k)$ at all. Which raises the following queries.
Queries.
1) In this specific case, would it constitute a flaw in my induction strategy if I showed that $y_{k+1} \geq y_{k+2}$ without explicitly making use of the assumption that $S(k)$ is true? In the sense that I assume the truth of $S(k)$ as a formality, but don't explicitly use it to deduce the truth of $S(k+1)$.
2) If there is no need to assume $S(k)$ is true to deduce the truth of $S(k+1)$, then surely induction is inappropriate in the sense that this hints at a more direct proof?
3) Or is the fact that a sequence of suprema $(y_n)$ is decreasing for a bounded sequence $(a_n)$ something so obvious from applying definitions that it shouldn't need proof at all?
4) The last possibility I have in mind is that my previous questions are a reflection of the fact that I do not know how to explicitly articulate how $y_{k} \geq y_{k+1}$ can be used to deduce $y_{k+1} \geq y_{k+2}$. If so, how do I explicitly articulate it? See related threads below.
Previous questions.
This questions here concerns exactly the same exercise for solution verification, and the OP writes:
Assume that $y_k \geq y_{k+1}$. Then, $y_{k+1}=\sup\{a_k : k + 1 \geq n\}= \max\{a_{k+1}, y_{k+2} \} ≥ y_{k+2}.$
Whilst this intuitively seems obvious, to me it feels unnatural. But perhaps that's because I am having difficulty reconciling my intuitions with why this must be true formally. I would appreciate some explanation on this.
Best Answer