Is it normal if proofs dont help me understand a concept better?

Question

I heard from people that if you learn proofs you will understand the concept better. But for me right now, i do not feel like thats true at all. In fact proofs sometimes confuse me more.

These are some examples: - Every bounded monotonic sequence is convergent - Fundamental theorem of calculus part 1 and 2 - If a series is convergent then the limit as n goes to infinity of an is 0

Is learning something from proofs something that develops over time or is what I say true in general? What is the value in learning proofs?

Answer 1

I feel that proofs usually help me better understand concepts. But your question made me mull over this, and I now have another take (hopefully more refined). Proofs help me understand the interactions or relations among concepts.

Take one of your examples, that the terms of a convergent series tend to zero. Just the statement of the theorem prompts one to wonder about the converse: is the condition sufficient as well as necessary? As you surely know, it isn't, with the harmonic series being the poster-child of counter-examples. I believe this fact deepens my understanding of the concept "convergent series".

Next, the proof. Briefly: the partial sums $s_n$ have to approach a limit, so $s_{n+1}$ and $s_n$ must both be within $\epsilon$ of that limit, and so must be within $2\epsilon$ of each other. In other words, $s_{n+1}-s_n$, which is just $a_n$, must be less than $2\epsilon$ in absolute value. Contemplating the proof, I picture the case where $a_n$ does not tend to 0. I see the partial sums jumping around like an excited puppy, unable to settle down to one spot. The proof focuses my attention on the significance of the subsidiary concept "partial sum". The proof also uses the triangle inequality---that's what allows us to conclude that since $s_n$ and $s_{n+1}$ are close to a third number, they are close to each other. This is a basic technique in analysis, and occurs over and over again.

The triangle inequality is also (arguably) the key property of metric spaces. The proof reveals the relation to another theorem: a convergent sequence of points in a metric space satisfies the Cauchy convergence criterion. Asking about the sufficiency of this condition leads to the notion of completeness. Also, the concept of a Cauchy sequence lies behind Cantor's construction of the reals from the rational, which generalizes to the idea of the completion of a metric space.

In the real numbers (or more generally any complete metric space), satisfying the Caucy convergence criterion is equivalent to being convergent. But we saw "convergent series" and "series whose terms approach 0" are not equivalent. Why not? Because our initial proof only looked at $s_{n+1}-s_n$, not $s_m-s_n$ as both $m$ and $n$ get large. So the proof sheds light on why the Cauchy convergence criterion takes the form it does.

Thus we see a web of connected ideas starting to grow. To my mind, seeing the placement of our initial theorem in this larger context enhances ones understanding of all the ideas in the web.

Answer 2

For many students, I think when learning proofs for the first time, proofs are not likely to elucidate a concept. Most beginning proof students are lost in the theorems, the new requirements for formality, strictness and requirement to cover what feels like every tiny little excruciating detail.

One goal of learning proofs is to get more familiar with the related theorems, their applications and how they relate to each other. This includes regularly asking oneself, will theorem A take me a step closer to my goal, what are the criteria to apply theorem A? How can I meet those criteria?

I think being able to prove something requires an intimate knowledge of the particular theorem to prove and other related theorems. Far more than simply applying the theorem.

Also, what is your long term goal? The theorem * Every bounded monotonic sequence is convergent* is usually taught in Real Analysis, a senior level course for math majors. For math majors, proofs are a foundational topic for many upper level courses. Graduate school in math depending on the course may be entirely proofs.

For engineers, proofs are ultimately much less important, except to develop critical thinking skills.