- #1

Mr Davis 97

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## Homework Statement

Prove rigorously that ##\displaystyle \lim \frac{n}{n^2 + 1} = 0##.

## Homework Equations

A sequence ##(s_n)## converges to ##s## if ##\forall \epsilon > 0 \exists N \in \mathbb{N} \forall n \in \mathbb{N} (n> N \implies |s_n - s| < \epsilon)##

## The Attempt at a Solution

Let ##\epsilon > 0##. Let ##\displaystyle N = \frac{1}{\epsilon}##. Let ##n \in \mathbb{N}##.

Then, if ##n > N##, we have that ##\displaystyle n > \frac{1}{\epsilon}## and so ##\displaystyle \frac{1}{n} < \epsilon##. Therefore, ##\displaystyle |\frac{n}{n^2+1} - 0| = \frac{n}{n^2+1} < \frac{n}{n^2} = \frac{1}{n} < \epsilon##. This proves that ##\displaystyle \lim \frac{n}{n^2 + 1} = 0##.

Is this a correct convergence proof?