Bounded For Sequences

Bounded (sequences)

A Sequence $(a_{n})$ is bounded if $\exists M > 0$ such that $M \in R$ and:

| a_{n} | < M

for all $n \in N$ .

Note that this mean ${a_{n} | n \in N}$ is bounded.

Theorem

A convergent sequence is bounded.

Proof
Let $a_{n} \to a$ . We need to show our $M$ . Notice that since $a_{n} \to a$ then use $ε = 1 > 0$ to get that $\exists N$ such that for all $n \geq N$ we have:

| a_{n} - a | < 1

Notice (as described in the figure) that:

| a_{n} | < | a | + 1

for every $n \geq N$ . This algebraically comes from that:

\begin{array}{r} - 1 < a_{n} - a < 1 \\ a - 1 < a_{n} < a + 1 \\ a_{n} < a + 1 \leq | a | + 1 \\ \Rightarrow | a_{n} | < | a | + 1 \end{array}

Now then choose $M = max (| a | + 1, | a_{1} |, . . ., | a_{N - 1} |) \in R$ . Then:

| a_{n} | < | a | + 1 = M

when $n \geq N$ . For when $n < N$ then by construction then our $| a_{n} | \leq M$ as desired.

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