# A Small Question about Rudin's Gap Construction [duplicate]

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In Rudin's PMA, he shows the following.

$A=$ { $p \in R^+ | p^2<2$}, B= { $p \in R^+ | p^2>2$}

Let $q = p - \frac{p^2 -2}{p+2}= \frac{2p+2}{p+2}$, then $q^2 -2= \frac{2(p^2- 2)}{(p+2)^2}$

Therefore, if $p$ is in $A$, then $q>p$ and $q$ is in A. If $p$ is in $B$, $0<q<p$ and $q$ is in $B$.

I do understand all of this, but I'm just wondering if there is any fundamental mathematical reasoning to construct a gap like this. It just seems to me that he found somehow that magical number $q$: I want more insight to see how he was able to find and construct it.

## marked as duplicate by Daniel Fischer real-analysis StackExchange.ready(function() { if (StackExchange.options.isMobile) return; $('.dupe-hammer-message-hover:not(.hover-bound)').each(function() { var$hover = $(this).addClass('hover-bound'),$msg = $hover.siblings('.dupe-hammer-message');$hover.hover( function() { $hover.showInfoMessage('', { messageElement:$msg.clone().show(), transient: false, position: { my: 'bottom left', at: 'top center', offsetTop: -7 }, dismissable: false, relativeToBody: true }); }, function() { StackExchange.helpers.removeMessages(); } ); }); }); Sep 5 '14 at 14:19

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