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{{space|5}}Let the smaller of the two numbers ω<sub>κ<sub>1</sub></sub>, ω<sub>κ<sub>2</sub></sub> be denoted by α', the larger by β'. (Their equality is impossible because we assumed that our sequence consists of nothing but unequal numbers.)

{{space|5}}Then according to the definition:<br>
{{space|12}}α&nbsp;<&nbsp;α'&nbsp;<&nbsp;β'&nbsp;<&nbsp;β{{space|hair}},<br>
furthermore:<br>                           
{{space|17}}κ<sub>1</sub>&nbsp;<&nbsp;κ<sub>2</sub>{{space|hair}};<br>
and all numbers ω<sub>μ</sub> of our sequence, 
for which μ ≤ κ<sub>2</sub>, do ''not'' lie in the interior of the interval [α',&nbsp;β'], as is immediately clear from the definition of the numbers κ<sub>1</sub>, κ<sub>2</sub>. Similarly, let ω<sub>κ<sub>3</sub></sub> and ω<sub>κ<sub>4</sub></sub> be the two numbers of our sequence with smallest indices that fall in the ''interior'' of the interval [α',&nbsp;β'] and let the smaller of the numbers ω<sub>κ<sub>3</sub></sub>, ω<sub>κ<sub>4</sub></sub> be denoted by α<nowiki>''</nowiki>, the larger by β<nowiki>''</nowiki>.

{{space|5}}Then one has: <br> 
{{space|12}}α'&nbsp;< α<nowiki>''</nowiki>&nbsp;<&nbsp;β<nowiki>''</nowiki>&nbsp;<&nbsp;β'{{space|hair}},<br>
{{space|14}}κ<sub>2</sub>&nbsp;<&nbsp;κ<sub>3</sub>&nbsp;<&nbsp;κ<sub>4</sub>{{space|hair}};<br>
and one sees that all numbers ω<sub>μ</sub> of our sequence, for which μ&nbsp;≤&nbsp;κ<sub>4</sub>, do ''not'' fall into the ''interior'' of the interval [α<nowiki>''</nowiki>,&nbsp;β<nowiki>''</nowiki>].

{{space|5}} After one has followed this rule to reach an interval {{nowrap|[α<sup>(ν - 1)</sup>, β<sup>(ν - 1)</sup>]}}, the next interval is produced by selecting the first two (i. e. with lowest indices) numbers of our sequence (ω) (let them be {{nowrap|ω<sub>κ<sub>2ν - 1</sub></sub>}} and ω<sub>κ<sub>2ν</sub></sub>) that fall into the ''interior'' of {{nowrap|[α<sup>(ν - 1)</sup>,&nbsp;β<sup>(ν - 1)</sup>]}}. Let the smaller of these two numbers be denoted  by α<sup>(ν)</sup>, the larger by β<sup>(ν)</sup>.

{{space|5}}The interval [α<sup>(ν)</sup>,&nbsp;β<sup>(ν)</sup>] then lies in the ''interior'' of all preceding intervals and has the ''specific'' relation with our sequence (ω) that all numbers ω<sub>μ</sub>, for which μ ≤ κ<sub>2ν</sub>, ''definitely do not lie in its interior''. Since obviously:<br>
{{space|12}}{{nowrap|κ<sub>1</sub> < κ<sub>2</sub> < κ<sub>3</sub> <  . . . , ω<sub>κ<sub>2ν – 2</sub></sub> < ω<sub>κ<sub>2ν – 1</sub></sub> < ω<sub>κ<sub>2ν</sub></sub>{{space|hair}}, . . . }}<br>
and these numbers, as indices, are ''whole'' numbers, so:<br>
{{space|12}}κ<sub>2ν</sub>&nbsp;≥&nbsp;2ν{{space|hair}},<br>
and hence:<br>
{{space|12}}ν&nbsp;<&nbsp;κ<sub>2ν</sub>{{space|hair}};<br>
thus, we can certainly say (and this is sufficient for the following):

{{space|5}}''That if ν is an arbitrary whole number, the [real] quantity ω<sub>ν</sub> lies outside the interval [α<sup>(ν)</sup>&nbsp;.&nbsp;.&nbsp;.&nbsp;β<sup>(ν)</sup>].''

Cantor's new proof first handles the easy case of the sequence ''P'' not being dense in the interval. Then it deals with the more difficult case of ''P'' being dense. This division into cases not only indicates which sequences are most difficult to handle, but it also reveals the important role denseness plays in the proof.<ref group=proof name=p />

In the first case, ''P'' is not dense in [''a'',&nbsp;''b'']. By definition, ''P'' is dense in [''a'',&nbsp;''b''] if and only if for all subintervals (''c'',&nbsp;''d'') of [''a'',&nbsp;''b''], there is an ''x''&nbsp;∈&nbsp;''P'' such that {{nowrap|''x'' ∈ (''c'', ''d'')}}. Taking the negation of each side of the "if and only if" produces: ''P'' is not dense in [''a'',&nbsp;''b''] if and only if there exists a subinterval (''c'',&nbsp;''d'') of [''a'',&nbsp;''b''] such that for all ''x''&nbsp;∈&nbsp;''P''{{space|hair|3}}: {{nowrap|''x'' ∉ (''c'', ''d'')}}. Therefore, every number in (''c'',&nbsp;''d'') is not contained in the sequence ''P''.<ref group=proof name=p /> This case handles [[#Case1|case 1]] and [[#Case3|case 3]] of Cantor's 1874 proof.

In the second case, ''P'' is dense in [''a'',&nbsp;''b'']. The denseness of ''P'' is used to [[recursively define]] a nested sequence of intervals that excludes all elements of ''P''. The definition begins with
''a''<sub>1</sub>&nbsp;=&nbsp;''a'' and ''b''<sub>1</sub>&nbsp;=&nbsp;''b''. The definition's inductive case starts with the interval (''a''<sub>''n''</sub>,&nbsp;''b''<sub>''n''</sub>), which because of the denseness of ''P'' contains infinitely many elements of ''P''. From these elements of ''P'', the two with smallest indices are used to define ''a''<sub>''n''&nbsp;+&nbsp;1</sub> and ''b''<sub>''n''&nbsp;+&nbsp;1</sub>; namely, ''a''<sub>''n''&nbsp;+&nbsp;1</sub> is the least of these two numbers and  ''b''<sub>''n''&nbsp;+&nbsp;1</sub> is the greatest. [[#x_n not in|Cantor's 1874 proof]] demonstrates that for all&nbsp;''n''{{space|hair}}:{{space|hair}} {{nowrap|''x''<sub>''n''</sub> ∉ (''a''<sub>''n''</sub>,&nbsp;''b''<sub>''n''</sub>)}}. The sequence ''a''<sub>''n''</sub> is increasing and bounded above by ''b'', so it has a limit ''a''<sub>∞</sub>, which satisfies ''a''<sub>''n''</sub>&nbsp;<&nbsp;''a''<sub>∞</sub>. The sequence ''b''<sub>''n''</sub> is decreasing and bounded below by ''a'', so it has a limit ''b''<sub>∞</sub>, which satisfies ''b''<sub>∞</sub>&nbsp;<&nbsp;''b''<sub>''n''</sub>. Also, ''a''<sub>''n''</sub>&nbsp;<&nbsp;''b''<sub>''n''</sub> implies ''a''<sub>∞</sub>&nbsp;≤&nbsp;''b''<sub>∞</sub>. Therefore, {{nowrap|''a''<sub>''n''</sub> < ''a''<sub>∞</sub> ≤ ''b''<sub>∞</sub> < ''b''<sub>''n''</sub>}}. If ''a''<sub>∞</sub>&nbsp;<&nbsp;''b''<sub>∞</sub>, then for every ''n'': ''x''<sub>''n''</sub>&nbsp;∉&nbsp;(''a''<sub>∞</sub>,&nbsp;''b''<sub>∞</sub>) because ''x''<sub>''n''</sub> is not in the larger interval (''a''<sub>''n''</sub>,&nbsp;''b''<sub>''n''</sub>). This contradicts ''P'' being dense in [''a'',&nbsp;''b'']. Therefore, ''a''<sub>∞</sub>&nbsp;=&nbsp;''b''<sub>∞</sub>. Since for all ''n'': {{nowrap|''a''<sub>∞</sub> ∈ (''a''<sub>''n''</sub>,&nbsp;''b''<sub>''n''</sub>)}} but {{nowrap|''x''<sub>''n''</sub> ∉ (''a''<sub>''n''</sub>,&nbsp;''b''<sub>''n''</sub>)}}, the limit ''a''<sub>∞</sub> is a real number that is not contained in the sequence ''P''.<ref group=proof name=p /> This case handles [[#Case2|case 2]] of Cantor's 1874 proof.

== Fraktur ==
<math>\qquad\quad\;\,\textbf{Fraktur}</math>{{break|2}}
<math>\qquad\quad\text{A}\quad\;\;\mathfrak{A}\quad\mathfrak{a}</math>{{break|2}}
<math>\qquad\quad\text{B}\quad\;\;\mathfrak{B}\quad\mathfrak{b}</math>{{break|2}}
<math>\qquad\quad\text{C}\quad\;\;\mathfrak{C}\quad\mathfrak{c}</math>{{break|2}}
<math>\qquad\quad\text{D}\quad\;\;\mathfrak{D}\quad\mathfrak{d}</math>{{break|2}}
<math>\qquad\quad\text{E}\quad\;\;\mathfrak{E}\quad\mathfrak{e}</math>{{break|2}}
<math>\qquad\quad\text{F}\quad\;\;\mathfrak{F}\quad\mathfrak{f}</math>{{break|2}}
<math>\qquad\quad\text{G}\quad\;\;\mathfrak{G}\quad\mathfrak{g}</math>{{break|2}}
<math>\qquad\quad\text{H}\quad\;\;\mathfrak{H}\quad\mathfrak{h}</math>{{break|2}}
<math>\qquad\quad\text{I}\quad\;\;\;\,\mathfrak{I}\quad\;\mathfrak{i}</math>{{break|2}}
<math>\qquad\quad\text{J}\quad\;\;\;\,\mathfrak{J}\quad\;\mathfrak{j}</math>{{break|2}}
<math>\qquad\quad\text{K}\quad\;\;\;\mathfrak{K}\quad\;\mathfrak{k}</math>{{break|2}}
<math>\qquad\quad\text{L}\quad\;\;\;\,\mathfrak{L}\quad\;\mathfrak{l}</math>{{break|2}}
<math>\qquad\quad\text{M}\quad\;\;\mathfrak{M}\quad\mathfrak{m}</math>{{break|2}}
<math>\qquad\quad\text{N}\quad\;\;\mathfrak{N}\quad\mathfrak{n}</math>{{break|2}}
<math>\qquad\quad\text{O}\quad\;\;\mathfrak{O}\quad\mathfrak{o}</math>{{break|2}}
<math>\qquad\quad\text{P}\quad\;\;\mathfrak{P}\quad\mathfrak{p}</math>{{break|2}}
<math>\qquad\quad\text{Q}\quad\;\;\mathfrak{Q}\quad\mathfrak{q}</math>{{break|2}}
<math>\qquad\quad\text{R}\quad\;\;\mathfrak{R}\quad\mathfrak{r}</math>{{break|2}}
<math>\qquad\quad\text{S}\quad\;\;\mathfrak{S}\quad\mathfrak{s}</math>{{break|2}}
<math>\qquad\quad\text{T}\quad\;\;\mathfrak{T}\quad\mathfrak{t}</math>{{break|2}}
<math>\qquad\quad\text{U}\quad\;\;\mathfrak{U}\quad\mathfrak{u}</math>{{break|2}}
<math>\qquad\quad\text{V}\quad\;\;\mathfrak{V}\quad\mathfrak{v}</math>{{break|2}}
<math>\qquad\quad\text{W}\quad\;\;\!\mathfrak{W}\quad\mathfrak{w}</math>{{break|2}}
<math>\qquad\quad\text{X}\quad\;\;\mathfrak{X}\quad\mathfrak{x}</math>{{break|2}}
<math>\qquad\quad\text{Y}\quad\;\;\mathfrak{Y}\quad\mathfrak{y}</math>{{break|2}}
<math>\qquad\quad\text{Z}\quad\;\;\mathfrak{Z}\quad\;\mathfrak{z}</math>{{break|2}}