How can we form the relation of inequality?

Question

If $r$ is irrational, there is a unique integer $n$ with $|r- n|<1/2$

There are $$|n−m|\le |r−n|+|r−m|< \frac12 + \frac12 =1$$ in the answer, I understood all the steps except the relation of $|n−m|\le$ , how can we know $|n−m|\le|r−n|+|r−m|$

Our information:

$|r−n|<1/2$
$|r−m|<1/2$
$|r−n|+|r−m|<1/2+1/2$

So how can we get $|n−m| \le |r−n|+|r−m|<1/2+1/2$

Thanks

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score 0 · Accepted Answer · answered Jan 21 '20 at 13:35

0

Key step is a variant of triangle inequality $|a-b| \le |a| + |b|$. You have $$ |n-m| = |(n-r) - (m-r)| \le |n-r| + |m-r| = |r-n|+|r-m| $$

answered Jan 21 '20 at 13:35

gt6989b

54,422

How can we form the relation of inequality?

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