Prove if { a n } → ∞, then { a n } is not bounded above. Give an indirect proof.

Suppose there is some M ≥ 0 such that | a n | ≤ M for all n ≥ 1. But by assumption there is some N ≥ 1 such that a n &gt; M for all n ≥ N, a contradiction.

Since a n → ∞, by definition, for all k ∈ N , there exist a n k such that a n k &gt; k and thus { a n } is not bounded above.

Suppose there is some M ≥ 0 such that | a n | ≤ M for all n ≥ 1. But by assumption there is some N ≥ 1 such that a n &gt; M for all n ≥ N, a contradiction.

Since a n → ∞, by definition, for all k ∈ N , there exist a n k such that a n k &gt; k and thus { a n } is not bounded above.

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High School
Geometry
High school geometry
Indirect Proof

Brandon Monroe

Open question

2022-08-16

Prove if

{a_{n}}

\to

\infty

, then

{a_{n}}

is not bounded above. Give an indirect proof.

Answer & Explanation

Jason Petersen

Beginner2022-08-17Added 13 answers

Suppose there is some

M \geq 0

such that

| a_{n} | \leq M

for all

n \geq 1

. But by assumption there is some

N \geq 1

such that

a_{n} > M

for all

n \geq N

, a contradiction.

Rose Graves

Beginner2022-08-18Added 3 answers

Since

a_{n} \to \infty

, by definition, for all

k \in N

, there exist a

n_{k}

such that

a_{n_{k}} > k

and thus

{a_{n}}

is not bounded above.

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