The Algebraic Limit Theorem (iii)
- \(\lim (ca_n) = ca\) for all \(c \in \mathbf{R}\);
- \(\lim (a_n + b_n) = a + b\);
- \(\lim (a_nb_n) = ab\);
- \(\lim (a_n/b_n) = a\);
For the definitions of sequences and what it means to for a sequence to converge, see this, for subsequences see this.
For the “show the limit” template and an example, see this.
Problem Discussion
We’ll follow a similar approach to proving (i) and (ii) in here and here For (iii) we want to show that \(\lim (a_nb_n) = ab\). So we want to find \(N\) such that for any \(\epsilon > 0\),
The trick here to add and subtract a term \(ab_n\) and then use the triangle inequality:
Using the same trick from (ii), we can make \(\lvert a_n - a \rvert\) be as small as we want and similarly we can make \(\lvert b_n - b \rvert\) be as small as we want. The idea is to bound these two terms to values such that the total will be exactly \(\epsilon\) which is what we want to prove. So, given that \(\lim b_n = b\), then we know that for some \(n \geq N_1\),
Similarly, given that \(\lim b_n = b\), then we know that for some \(n \geq N_2\),
We still need to bound the term \(|b_n|\) since it’s not a constant. One trick is to use the convergent sequences theorem which says that all convergent sequences are bounded by some term \(M\). This means that we can replace the bound above with the following:
If we plug in everything back, we get
Finally we can choose \(N\) to be the maximum of \(N_1\) and \(N_2\) to guarantee the above bounds.
Formal Proof
Let \(\epsilon > 0\) be arbitrary. We are given that \(\lim a_n = a\). This means that for some \(n \geq N_1\) and for any choice of \(\epsilon\) , the following holds:
Since this holds for any \(\epsilon\), let \(\epsilon = \big(\frac{1}{|a|}\frac{\epsilon}{2}\big)\) so,
Similarly, given that \(\lim b_n = b\). This means that for some \(n \geq N_2\), the following holds:
\(M\) is an upper bound on the sequence \((b_n)\) guaranteed by convergent sequences theorem. Therefore, choose a natural number \(N\) satisfying \(\max(N_1,N_2)\). We now verify that this choice has the desired property. Let \(n \geq N\). Then,
as we wanted to show. \(\blacksquare\)
References: