what is the difference between bounded and convergent?

Question

I know that bounded means to have an upper or lower bound.

Let $E \subset \mathbb{R}$ be nonempty.

1. The set $E$ is said to be bounded above if and only if there is an $M \in \mathbb{R}$ such that $a \leq M$ for all $a \in E$, in which case $M$ is called an upper bound of $E$.

2. A number $s$ is called a supremum of the set $E$ if and only if $s$ is an upper bbound of $E$ and $s \leq M$ for all upper bounds $M$ of $E$ (In this case we shall say that $E$ has a finite supremum $s$ and write $s=\sup E$.

Let $E \subset \mathbb{R}$ be nonempty.

1. the set $E$ is said to be bounded below if and only if there is an $m \in \mathbb{R}$ such that $a \geq m$ for all $a\in E$, in which case $m$ is called a lower bound of the set $E$.

2. A number $t$ is called an infimum of the set $E$ if and only if $t$ is a lower bound of $E$ and $t \geq m$ for all lower bounds $m$ of $E$. In this case we shall say that $E$ has an infimum $t$ and write $t = \inf E$

3. $E$ is said to be bounded if and only if it is bounded both above and below.

The meaning of convergence

A sequence of real number $\{ x_n \}$ is said to converge to a real number $a \in \mathbb{R}$ if and only if for every $\epsilon > 0$ there is an $N \in \mathbb{N}$ (which in general depends on $\epsilon$) such that $$ n \geq N \mbox{ implies } \vert x_n - a \vert < \epsilon$$

In my previous classes, it was taught to take the the limit of the function or series to find the value which the function or series converges at the value. Is the value a function or series converges at the $\sup $ or $\inf$?

Answer 1

I think there's two points to address here. The first is that a sequence can be bounded from both above and below, yet fail to be convergent. Consider the sequence $$\{x_n\} = \{(-1)^n\} = -1,1,-1,1, \cdots$$ Here we have $\displaystyle\sup_{n} \{x_n\} = 1$, $\displaystyle\inf_{n} \{x_n\} = -1$, and $\{x_n\}$ does not converge (if you pick say $\epsilon = 0.1$, then there are infinitely many $n$ for which $|x_n - 1| > 0.1$ and for which $|x_n - (-1)| > 0.1$).

Furthermore, when a sequence $\{x_n\}$ does converge, the limit of a sequence need not equal either $\displaystyle\sup_n\{x_n\}$ or $\displaystyle\inf_n\{x_n\}$. For example, we see with the sequence $\{x_n\} = \{\frac{(-1)^n}{n}\}$ that $$\displaystyle\sup_n\{x_n\} = \frac{1}{2}, \: \:\displaystyle\inf_n\{x_n\} = -1, \: \: \displaystyle\lim_{n \to \infty} \{x_n\} = 0$$

In that last example, I kind of cheated, by picking a sequence where the inf and sup of the elements of the sequence were the first two, and the convergence of a sequence talks about the behavior at the tail of the sequence. When we want to talk about upper and lower bounds of sequences, we usually like to talk about the limit superior, or $$\displaystyle\limsup_{n \to \infty} \{x_n\} = \displaystyle\lim_{n \to \infty} \left(\sup_{N} \{x_N | \: N \geq n \} \right)$$ and the limit inferior, or $$\displaystyle\liminf_{n \to \infty} \{x_n\} = \displaystyle\lim_{n \to \infty} \left(\inf_{N} \{x_N | \: N \geq n \} \right)$$ This let's you bound the values of $x_n$ for arbitrarily large $n$, while allowing for a few badly behaved values of $x_n$ for small $n$. Then, we say that a sequence $\{x_n\}$ approaches a finite limit $M$ if and only iff $$ \displaystyle\limsup_{n \to \infty} \{x_n\} = \displaystyle\liminf_{n \to \infty} \{x_n\} = M$$ and you can read about the proof of this result here (Proof that a sequence converges to a finite limit iff lim inf equals lim sup).