Can you find a convergent sequence that is not bounded?
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In fact, it takes not much to see convergence of a real sequence implies boundedness of the sequence; let $(x_{n})$ be a sequence converging to some $l \in \mathbb{R}$. Then there is some $N \geq 1$ such that $n \geq N$ only if $|x_{n} - l| < 1$, and hence $|x_{n}| < 1 + |l|$ for all $n \geq N$ by "triangle inequality"; then the number $\max \{ \max_{1 \leq n \leq N-1}|x_{n}|, 1 + |l| \}$ is a bound for the sequence $(x_{n})$.
The above argument can generalize to any sequence in any space on which a metric is definable.
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Betelhem Dessie
Updated on August 14, 2022Comments
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Betelhem Dessie about 1 year
Can you find a convergent sequence that is not bounded? Examples please
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Sammy Black about 8 yearsWhat does convergent mean? What does bounded mean? Write these down first.
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mod0 about 5 yearsI have a question about the above explanation, what if any of the $x_n$ for $n = 1 ... N - 1$ are infinity? As an example, consider the sequence $(x_n) = \frac{1}{n - 1}$, the sequence converges to zero as $n \rightarrow \infty$, but $x_1 = \infty$. In this case, the sequence is convergent, but is not bounded.