Limit of product of sequences is the product of the limits of the sequences

I want to prove that if:

$$\lim_{n \to \infty}s_n = L_1, \lim_{n \to \infty}t_n = L_2$$

then $$\lim_{n \to \infty}(s_n t_n) = L_1L_2$$

where $s_n, t_n$ are complex sequences (I'm working through Rudin's baby rudin, and I did this proof slightly different than him, so I would like it if someone took the time to look through my proof and verify whether it's correct)

My attempt (scratch proof: finding the appropriate epsilons etc):

By definition of limit, there are positive integers $N_1,N_2$ such that:

$$n >N_1 \Rightarrow |s_n - L_1| <\epsilon'$$ $$n >N_2 \Rightarrow |t_n - L_2| <\epsilon''$$

where we can freely choose $\epsilon', \epsilon'' >0$

Let $\epsilon > 0$

Now, if $n > \max\{N_1,N_2\}$, then:

$$|s_nt_n - L_1L_2| = |s_n(t_n - L_2) + s_nL_2 - L_1L_2|$$ $$\leq |s_n||t_n - L_2| + |L_2||s_n - L_1|$$ $$\leq |s_n|\epsilon'' + |L_2|\epsilon'$$

We want this expression to be smaller than $\epsilon$, but we can't make our $\epsilon$ depend on $n$, therefore we have to find an upper bound for $|s_n|$

There is a positive integer $N_3$ such that:

$$n > N_3 \Rightarrow |s_n - L_1| < 1$$ $$\Rightarrow |s_n| < 1 + |L_1|$$

so for $n > \max\{N_1,N_2,N_3\}$, we have:

$$|s_n|\epsilon'' + |L_2|\epsilon' < (1 + |L_1|)\epsilon'' + |L_2|\epsilon'$$

and by chosing $\epsilon'' = \epsilon' = \frac{\epsilon}{1+|L_1|+|L_2|}$, the expression becomes smaller than $\epsilon$

Rigorous proof

Let $\epsilon > 0$

By definition of limit, there are positive integers $N_1,N_2, N_3$ such that:

$$n >N_1 \Rightarrow |s_n - L_1| <\frac{\epsilon}{1+|L_1|+|L_2|}$$ $$n >N_2 \Rightarrow |t_n - L_2| <\frac{\epsilon}{1+|L_1|+|L_2|}$$ $$n > N_3 \Rightarrow |s_n - L_1| < 1 \Rightarrow |s_n| < 1 + |L_1|$$

and for $n > \max\{N_1,N_2,N_3\}$, we have:

$$|s_nt_n - L_1L_2| = |s_n(t_n - L_2) + s_nL_2 - L_1L_2|$$ $$\leq |s_n||t_n - L_2| + |L_2||s_n - L_1|$$ $$< (1 + |L_1|)\frac{\epsilon}{1+|L_1|+|L_2|} + |L_2|\frac{\epsilon}{1+|L_1|+|L_2|} = \frac{\epsilon}{1+|L_1|+|L_2|}(1+ |L_1| + |L_2|) = \epsilon$$

Alternative proof:

Let $\epsilon > 0$

$(s_n)$ converges, so $(|s_n|)$ converges and it is bounded, meaning that there is a positive real number $S$ such that $|s_n| \leq S$ for every positive $n$.

By definition of limit, there are positive integers $N_1,N_2$ such that:

$$n >N_1 \Rightarrow |s_n - L_1| <\frac{\epsilon}{S + |L_2|}$$ $$n >N_2 \Rightarrow |t_n - L_2| <\frac{\epsilon}{S + |L_2|}$$

Now, for $n > \max\{N_1,N_2\}$, we have:

$$|s_nt_n - L_1L_2| = |s_n(t_n - L_2) + s_nL_2 - L_1L_2|$$ $$\leq |s_n||t_n - L_2| + |L_2||s_n - L_1|$$ $$< (S+|L_2|)\frac{\epsilon}{S + |L_2|} = \epsilon$$

Note that this wouldn't work whenever $S + |L_2| = 0$. However, if $S=0$, then $s_n$ is the null-sequence and the theorem is trivial, so we can safely assume $S \neq 0$ such that the denominator never becomes $0$.

QED

Is this correct?

Your “Rigorous proof” is not correct. Look at the way you chose $N_1$, $N_2$, and $N_3$. Only the sequence $(s_n)_{n\in\mathbb N}$ is mentioned, not the sequence $(t_n)_{n\in\mathbb N}$. You can't get a correct proof without using both of them.

Here is how I would prove the same statement. Please keep in mind that I always try to avoid to introduce asymmetries when the hypothesis are symmetric. In this case, what is supposed about both sequences is the same and therefore I will avoid to treat them in two different ways.

For each $n\in\mathbb N$, you know that$$s_nt_n-L_1L_2=(s_n-L_1)L_2+(t_n-L_2)L_1+(s_n-L_1)(t_n-L_2)$$and therefore that$$|s_nt_n-L_1L_2|\leqslant|s_n-L_1|.|L_2|+|t_n-L_2|.|L_1|+|s_n-L_1|.|t_n-L_2|.$$Let $\varepsilon>0$ and choose $N_1\in\mathbb N$ such that$$n\geqslant N_1\Longrightarrow|s_n-L_1|<\min\left\{\frac\varepsilon{3\bigl(|L_2|+1\bigr)},\sqrt{\frac\varepsilon3}\right\}.$$Also, choose $N_2\in\mathbb N$ such that$$n\geqslant N_2\Longrightarrow|t_n-L_2|<\min\left\{\frac\varepsilon{3\bigl(|L_1|+1\bigr)},\sqrt{\frac\varepsilon3}\right\}.$$Let $N=\max\{N_1,N_2\}$. Then, if $n\geqslant N$,\begin{multline*}|s_n-L_1|.|L_2|+|t_n-L_2|.|L_1|+|s_n-L_1|.|t_n-L_2|<\\<\frac{\varepsilon|L_2|}{3\bigl(|L_2|+1\bigr)}+\frac{\varepsilon|L_1|}{3\bigl(|L_1|+1\bigr)}+\frac\varepsilon3<\varepsilon.\end{multline*}

• Thanks for your answer. This was copy paste error. I edited it. Is it correct now? – user370967 Aug 10 '17 at 12:03
• @Math_QED It's better, but not correct. You used implicitly (I think) that $|s_n|<1+|L_1|+|L_2|$ if $n$ is large enough; you should be explicit. Besides, aftar the last $<$ sign there is a factor $2$ missing (you are adding a thing with itself), and therefore in the end what you actually get is $2\epsilon$, not $\epsilon$. – José Carlos Santos Aug 10 '17 at 12:10
• I used $|s_n| < 1 + |L_1|$ for $n$ large enough. I wrote this down now in the proof. Hope it's correct now. Don't know what factor $2$ I'm missing though. – user370967 Aug 10 '17 at 12:20
• @Math_QED It looks correct now. There is no factor $2$ missing after all; I had the impression that you were getting $(1+|L_1|+|L_2|)\frac\epsilon{1+|L_1|+|L_2|}$ twice, but I see now that that was not the case. Do you want me to add to my answer how I would have done it? – José Carlos Santos Aug 10 '17 at 12:23
• Thanks for all your help :). I think your proof is quite complicated by introducing minimums, but on the other hand you don't asymmetries. I prefer my proofs but that's just a matter of taste. – user370967 Aug 10 '17 at 14:56

hint

there is an error in line 37.

the easiest way is to use the fact that a convergent sequence is bounded.

$$|s_n|\le S$$

and $$|s_nt_n-L_1L_2|\le$$ $$S|t_n-L_2|+|L_2||s_n-L_1|\le$$ $$S\frac {\epsilon}{2S}+|L_2|\frac {\epsilon}{2|L_2|}$$

if $$SL_2\ne 0$$

If $L_2=0$ and $S\ne 0$

$$|s_nt_n|\le S\frac {\epsilon}{S}$$ for enough large $n$.

• This does not answer the OP's question, which is about verifying their proof. – Benjamin Dickman Aug 10 '17 at 11:51
• This is actually a useful answer. – user370967 Aug 10 '17 at 12:06
• So, as an alternative answer: $s_n$ converges, so $|s_n|$ converges, so $(|s_n|)$ (the sequence of moduli of the complex numbers) is bounded, meaning there is an element $S \in \mathbb{R}$ such that $|s_n| \leq S$ for all $n$ and I can use this in the proof? – user370967 Aug 10 '17 at 12:13
• This proof does not work if either $S = 0$ or $L_2 = 0$. I followed your suggestion and added my own proof. Could you maybe look at it? – user370967 Aug 10 '17 at 12:35