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I don't really understand how a null sequence is different to a cauchy sequence?

Could someone show me this with some examples please?

Thanks :)

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    By "null sequence", you mean "a sequence which converges to $0$"? – 2017-01-15
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    @OpenBall I'm not totally sure, here is the definition I have been given: "A null sequence is a sequence {an} of rational numbers with the following property. For any rational number ε > 0 there exists a natural number N such that if a natural number n ≥ N, then |an| < ε" – 2017-01-15
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    $a_n=1-1/n$ is an example. – 2017-01-15
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    Every convergent sequence is a Cauchy sequence. – 2017-01-15
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    @Spenser is that because the sequences gets larger and n increases or beacuse it doesn't tend towards 0? – 2017-01-15
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    Its because it doesn't tend towards $0$. – 2017-01-15

2 Answers 2

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Consider $$a_n=1+1/n.$$

This is Cauchy, but does not converge to zero.

In particular, using your definition from the comments, let $\epsilon=1$, to see that this sequence is not null.

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In the real numbers, a sequence converges if and only if it is a Cauchy sequence. Thus, a null sequence is a Cauchy sequence that converges to $0$. But there are Cauchy sequences which converge to other values.

For example if $L\in\mathbb{R}$ is fixed then $$a_n=L+\frac{1}{n}$$ is a Cauchy sequence that converges to $L$. The important point is that limits are unique, i.e. a sequence cannot converge to two distinct numbers. Thus, $a_n$ is a null sequence if and only if $L=0$.