Exercise 5.3.1

Exercise statement

Prove Proposition 5.3.3.

Proposition 5.3.3 (Formal limits are well-defined). Let $x = \mathrm{LIM}_{n \to \infty} a_n$ , $y = \mathrm{LIM}_{n \to \infty} b_n$ , and $z = \mathrm{LIM}_{n \to \infty} c_n$ be real numbers. Then, with the above definition of equality for real numbers, we have $x=x$ . Also, if $x=y$ , then $y=x$ . Finally, if $x=y$ and $y=z$ , then $x=z$ .

Hints

You may find Proposition 4.3.7 to be useful.

How to think about the exercise

This is a straightforward exercise, so I don’t have anything to say.

Model solution

To show that $x=x$ we need to show that $(a_n)_{n=1}^\infty$ is equivalent to $(a_n)_{n=1}^\infty$ . So let $\varepsilon > 0$ . Then for $N:=1$ we have $|a_n - a_n| = 0 < \varepsilon$ for every $n\geq N$ , so $x=x$ as desired.

To show that $x=y$ implies $y=x$ , suppose $(a_n)_{n=1}^\infty$ and $(b_n)_{n=1}^\infty$ are equivalent. This means that for every $\varepsilon > 0$ there is some $N\geq1$ such that for all $n\geq N$ we have $|a_n-b_n| \leq \varepsilon$ . By Proposition 4.3.3(d) we know that $|a_n-b_n| = |b_n-a_n|$ , so the above is actually also saying that $(b_n)_{n=1}^\infty$ and $(a_n)_{n=1}^\infty$ are equivalent. More formally, let $\varepsilon > 0$ . Then by the equivalence of $(a_n)_{n=1}^\infty$ and $(b_n)_{n=1}^\infty$ we are given some $N\geq 1$ . Then for $n\geq N$ we have $|b_n-a_n| = |a_n-b_n| \leq \varepsilon$ .

Finally suppose $x=y$ and $y=z$ . This means that $(a_n)_{n=1}^\infty$ and $(b_n)_{n=1}^\infty$ are equivalent, and that $(b_n)_{n=1}^\infty$ and $(c_n)_{n=1}^\infty$ are equivalent. Let $\varepsilon >0$ be given. Then there exists $N_1 \geq 1$ such that $a_n$ is $\varepsilon/2$ -close to $b_n$ for all $n\geq N_1$ , and there exists $N_2\geq 1$ such that $b_n$ is $\varepsilon/2$ -close to $c_n$ for all $n\geq N_2$ . So if we pick $N:=\max(N_1,N_2)$ then for $n\geq N$ we have both $n\geq N_1$ and $n\geq N_2$ . Thus by Proposition 4.3.7(c) we see that $a_n$ and $c_n$ are $\varepsilon$ -close. Thus $x=z$ as required.