How We Got From There To Here: A Story of Real Analysis

As you’ve seen from the previous sections, the formal definition of the convergence of a sequence is meant to capture rigorously our intuitive understanding of convergence. However, the definition itself is an unwieldy tool. If only there was a way to be rigorous without having to run back to the definition each time. Fortunately, there is a way. If we can use the definition to prove some general rules about limits then we could use these rules whenever they applied and be assured that everything was still rigorous. A number of these should look familiar from calculus.

In proving the familiar limit theorems, the following will prove to be a very useful tool.

We will often informally state this theorem as “the limit of a sum is the sum of the limits.” However, to be absolutely precise, what it says is that if we already know that two sequences converge, then the sequence formed by summing the corresponding terms of those two sequences will converge and, in fact, converge to the sum of those individual limits. We’ll provide the scrapwork for the proof of this and leave the formal write-up as an exercise. Note the use of the triangle inequality in the proof.

We can make this whole thing less than \(\eps\text{,}\) provided we make each term in the sum less than \(\frac{\eps}{2}\text{.}\) We can make \(\big|b\big|\big|a_n-a\big|\lt \frac{\eps}{2}\) if we make \(\big|a_n-a\big|\lt \frac{\eps}{2|b|}\text{.}\) But wait! What if \(b=0\text{?}\) We could handle this as a separate case or we can do the following “slick trick.” Notice that we can add one more line to the above string of inequalities: \(\left|a_n\right|\left|b_n-b\right|+\left|b\right|\left|a_n-a\right|\lt \left|a_n \right|\left|b_n-b\right|+\left(\left|b\right|+1\right)\left|a_n-a \right|\text{.}\) Now we can make \(\left|a_n-a\right|\lt \frac{\eps}{2\left(|b|+1\right)}\) and not worry about dividing by zero.

Making \(\big|a_n\big|\big|b_n-b\big|\lt \frac{\eps}{2}\) requires a bit more finesse. At first glance, one would be tempted to try and make \(\big|b_n-b\big|\lt \frac{\eps}{2|a_n|}\text{.}\) Even if we ignore the fact that we could be dividing by zero (which we could handle), we have a bigger problem. According to the definition of \(\lim_{n\rightarrow\infty}b_n=b\text{,}\) we can make \(\big|b_n-b\big|\) smaller than any given fixed positive number, as long as we make \(n\) large enough (larger than some \(N\) which goes with a given epsilon). Unfortunately, \(\frac{\eps}{2|a_n|}\) is not fixed as it has the variable \(n\) in it; there is no reason to believe that a single \(N\) will work with all of these simultaneously. To handle this impasse, we need the following:

At this point, we should be able to make the last line of this less than \(\eps\text{.}\)

Just as Theorem 6.2.6 says that the limit of a product is the product of the limits, we can prove the analogue for quotients.

These theorems allow us to compute limits of complicated sequences and rigorously verify that these are, in fact, the correct limits without resorting to the definition of a limit.

There is one more tool that will prove to be valuable.

The Squeeze Theorem holds even if \(r_n\leq s_n\leq t_n\) holds for only sufficiently large \(n\text{;}\) i.e., for \(n\) larger than some fixed \(N_0\text{.}\) This is true because when you find an \(N_1\) that works in the original proof, this can be modified by choosing \(N=\)max\(\left(N_0,N_1\right)\text{.}\) Also note that this theorem really says two things: \(\left(s_n\right)\) converges and it converges to \(s\text{.}\) This subtle point affects how one should properly use the Squeeze Theorem.

Notice that this proof is completely rigorous. Also notice that this is the proper way to use the Squeeze Theorem. Here is an example of an {improper} use of the Squeeze Theorem.

This is incorrect in form because it presumes that \(\lim_{n\rightarrow\infty}\frac{n+1}{n^2}\) exists, which we don’t yet know. If we knew that the limit existed to begin with, then this would be fine. The Squeeze Theorem proves that the limit does in fact exist, but it must be so stated.

These general theorems will allow us to rigorously explore convergence of power series in the next chapter without having to appeal directly to the definition of convergence. However, you should remember that we used the definition to prove these results and there will be times when we will need to apply the definition directly. However, before we go into that, let’s examine divergence a bit more closely.