The sequence 1/n tends to zero

Builds on Limit of a sequence Epsilon-N definition of the limit of a sequence

Example

Let $a_n = 1/n$ for each integer $n \ge 1$ . Then $a_n \to 0$ as $n \to \infty$ . Let $\varepsilon > 0$ . By the Archimedean property of $\mathbb{R}$ , there is a natural number $N$ with $N > 1/\varepsilon$ . For each $n \ge N$ we get $|a_n - 0| = 1/n \le 1/N < \varepsilon$ . By the epsilon-N definition, this shows $1/n \to 0$ .

Remarks

The proof uses the Archimedean property, which says that the natural numbers are not bounded above in $\mathbb{R}$ . With the algebra of limits, it gives many more limits, for example $1/n^2 \to 0$ and $(n + 1)/n \to 1$ . Mathlib states it for $1/n$ over $\mathbb{R}$ as tendsto_one_div_atTop_nhds_zero_nat.

Reading path

To understand this page from the ground up, read these 5 pages in order. The foundations come first.

Show the path as a graph

Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

no evidence level

Sequence definition
Real sequence definition
Limit of a sequence definition
Epsilon-N definition of the limit of a sequence statement
The sequence 1/n tends to zero example

Tag	`08QAX4H1PF`
URI	`m2://08QAX4H1PF`
Kind	example
Writing	W0 stub
MSC	40A05
Instance of	Convergent sequence
Uses	Limit of a sequence, Epsilon-N definition of the limit of a sequence

The sequence 1/n tends to zero

Example

Remarks

Reading path

Further reading

What this page needs for W1

What links here