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Omni-MATH / Do there exist two bounded sequences a₁,a₂,… and b₁,b₂,… such that for each positive integers n and m>n at least one of the two inequalities …

Problem

Answer published by the source. Consult the official source to check your work against its answer.

problem

Do there exist two bounded sequences a1,a2,…a_{1}, a_{2}, \ldots and b1,b2,…b_{1}, b_{2}, \ldots such that for each positive integers nn and m>nm > n at least one of the two inequalities |am−an|>1n,|bm−bn|>1n|a_{m} - a_{n}| > \frac{1}{\sqrt{n}}, |b_{m} - b_{n}| > \frac{1}{\sqrt{n}} holds?
Plain-text mathematical notation (without MathML)
Do there exist two bounded sequences a₁,a₂,… and b₁,b₂,… such that for each positive integers n and m>n at least one of the two inequalities |a_(m)−a_(n)|>(1)/(√(n)),|b_(m)−b_(n)|>(1)/(√(n)) holds?
Original LaTeX notation
Do there exist two bounded sequences $a_{1}, a_{2}, \ldots$ and $b_{1}, b_{2}, \ldots$ such that for each positive integers $n$ and $m > n$ at least one of the two inequalities $|a_{m} - a_{n}| > \frac{1}{\sqrt{n}}, |b_{m} - b_{n}| > \frac{1}{\sqrt{n}}$ holds?

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Source and history

task_id
6ff1e3a0-b153-5f10-ac40-2faad9de01b0
task_key
test--6ff1e3a0-b153-5f10-ac40-2faad9de01b0
task_revision_id
1
config / split
/ test

Official source

initial import