Dini's theorem

<h2 id="formal-statement">Formal statement</h2>
<p>If 
  
    
      
        X
      
    
    {\displaystyle X}
  
 is a <a href="/facts/Compact_space/d0cgXJH7">compact</a> <a href="/facts/Topological_space/v2ut7CbK">topological space</a>, and 
  
    
      
        (
        
          f
          
            n
          
        
        
          )
          
            n
            ∈
            
              N
            
          
        
      
    
    {\displaystyle (f_{n})_{n\in \mathbb {N} }}
  
 is a <a href="/facts/Monotonically_increasing/MjQK0YL2">monotonically increasing</a> <a href="/facts/Sequence/gV2S4uqp">sequence</a> (meaning 
  
    
      
        
          f
          
            n
          
        
        (
        x
        )
        ≤
        
          f
          
            n
            +
            1
          
        
        (
        x
        )
      
    
    {\displaystyle f_{n}(x)\leq f_{n+1}(x)}
  
 for all 
  
    
      
        n
        ∈
        
          N
        
      
    
    {\displaystyle n\in \mathbb {N} }
  
 and 
  
    
      
        x
        ∈
        X
      
    
    {\displaystyle x\in X}
  
) of <a href="/facts/Continuous_function/sKbl02pB">continuous</a> <a href="/facts/Real-valued_function/jetqlJpK">real-valued functions</a> on 
  
    
      
        X
      
    
    {\displaystyle X}
  
 which converges <a href="/facts/Pointwise_convergence/TXGpeyIK">pointwise</a> to a continuous function 
  
    
      
        f
        :
        X
        →
        
          R
        
      
    
    {\displaystyle f\colon X\to \mathbb {R} }
  
, then the convergence is <a href="/facts/Uniform_convergence/ecz9eNWn">uniform</a>. The same conclusion holds if 
  
    
      
        (
        
          f
          
            n
          
        
        
          )
          
            n
            ∈
            
              N
            
          
        
      
    
    {\displaystyle (f_{n})_{n\in \mathbb {N} }}
  
 is monotonically decreasing instead of increasing. The theorem is named after <a href="/facts/Ulisse_Dini/N33MSsfn">Ulisse Dini</a>.<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a>
</p><p>This is one of the few situations in mathematics where pointwise convergence implies uniform convergence; the key is the greater control implied by the monotonicity. The limit function must be continuous, since a uniform limit of continuous functions is necessarily continuous. The continuity of the limit function cannot be inferred from the other hypothesis (consider 
  
    
      
        
          x
          
            n
          
        
      
    
    {\displaystyle x^{n}}
  
 in 
  
    
      
        [
        0
        ,
        1
        ]
      
    
    {\displaystyle [0,1]}
  
.)
</p>
<h2 id="proof">Proof</h2>
<p>Let 
  
    
      
        ε
        >
        0
      
    
    {\displaystyle \varepsilon >0}
  
 be given.  For each 
  
    
      
        n
        ∈
        
          N
        
      
    
    {\displaystyle n\in \mathbb {N} }
  
, let 
  
    
      
        
          g
          
            n
          
        
        =
        f
        −
        
          f
          
            n
          
        
      
    
    {\displaystyle g_{n}=f-f_{n}}
  
, and let 
  
    
      
        
          E
          
            n
          
        
      
    
    {\displaystyle E_{n}}
  
 be the set of those 
  
    
      
        x
        ∈
        X
      
    
    {\displaystyle x\in X}
  
 such that 
  
    
      
        
          g
          
            n
          
        
        (
        x
        )
        <
        ε
      
    
    {\displaystyle g_{n}(x)<\varepsilon }
  
. Each 
  
    
      
        
          g
          
            n
          
        
      
    
    {\displaystyle g_{n}}
  
 is continuous, and so each 
  
    
      
        
          E
          
            n
          
        
      
    
    {\displaystyle E_{n}}
  
 is open (because each 
  
    
      
        
          E
          
            n
          
        
      
    
    {\displaystyle E_{n}}
  
 is the <a href="/facts/Preimage/KANefMAl">preimage</a> of the open set 
  
    
      
        (
        −
        ∞
        ,
        ε
        )
      
    
    {\displaystyle (-\infty ,\varepsilon )}
  
 under 
  
    
      
        
          g
          
            n
          
        
      
    
    {\displaystyle g_{n}}
  
, a continuous function).  Since 
  
    
      
        (
        
          f
          
            n
          
        
        
          )
          
            n
            ∈
            
              N
            
          
        
      
    
    {\displaystyle (f_{n})_{n\in \mathbb {N} }}
  
 is monotonically increasing, 
  
    
      
        (
        
          g
          
            n
          
        
        
          )
          
            n
            ∈
            
              N
            
          
        
      
    
    {\displaystyle (g_{n})_{n\in \mathbb {N} }}
  
 is monotonically decreasing, it follows that the sequence 
  
    
      
        
          E
          
            n
          
        
      
    
    {\displaystyle E_{n}}
  
 is ascending (i.e. 
  
    
      
        
          E
          
            n
          
        
        ⊂
        
          E
          
            n
            +
            1
          
        
      
    
    {\displaystyle E_{n}\subset E_{n+1}}
  
 for all 
  
    
      
        n
        ∈
        
          N
        
      
    
    {\displaystyle n\in \mathbb {N} }
  
).  Since 
  
    
      
        (
        
          f
          
            n
          
        
        
          )
          
            n
            ∈
            
              N
            
          
        
      
    
    {\displaystyle (f_{n})_{n\in \mathbb {N} }}
  
 converges pointwise to 
  
    
      
        f
      
    
    {\displaystyle f}
  
, it follows that the collection 
  
    
      
        (
        
          E
          
            n
          
        
        
          )
          
            n
            ∈
            
              N
            
          
        
      
    
    {\displaystyle (E_{n})_{n\in \mathbb {N} }}
  
 is an <a href="/facts/Open_cover/2VUdelA4">open cover</a> of 
  
    
      
        X
      
    
    {\displaystyle X}
  
.  By compactness, there is a finite subcover, and since 
  
    
      
        
          E
          
            n
          
        
      
    
    {\displaystyle E_{n}}
  
 are ascending the largest of these is a cover too. Thus we obtain that there is some positive integer 
  
    
      
        N
      
    
    {\displaystyle N}
  
 such that  
  
    
      
        
          E
          
            N
          
        
        =
        X
      
    
    {\displaystyle E_{N}=X}
  
.  That is, if 
  
    
      
        n
        >
        N
      
    
    {\displaystyle n>N}
  
 and 
  
    
      
        x
      
    
    {\displaystyle x}
  
 is a point in 
  
    
      
        X
      
    
    {\displaystyle X}
  
, then 
  
    
      
        
          |
        
        f
        (
        x
        )
        −
        
          f
          
            n
          
        
        (
        x
        )
        
          |
        
        <
        ε
      
    
    {\displaystyle |f(x)-f_{n}(x)|<\varepsilon }
  
, as desired.
</p>
<h2 id="notes">Notes</h2>

<ul><li><a href="/facts/Robert_G._Bartle/QDhzW2vc">Bartle, Robert G.</a> and Sherbert Donald R.(2000) "Introduction to Real Analysis, Third Edition" Wiley. p 238. – Presents a proof using gauges.</li>
<li>Edwards, Charles Henry (1994) [1973]. <i>Advanced Calculus of Several Variables</i>. Mineola, New York: Dover Publications. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-486-68336-2.</li>
<li>Graves, Lawrence Murray (2009) [1946]. <i>The theory of functions of real variables</i>. Mineola, New York: Dover Publications. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-486-47434-2.</li>
<li><a href="/facts/Avner_Friedman/b0KkPJFq">Friedman, Avner</a> (2007) [1971]. <i>Advanced calculus</i>. Mineola, New York: Dover Publications. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-486-45795-6.</li>
<li><a href="/facts/J%25C3%25BCrgen_Jost/lRS3cQWE">Jost, Jürgen</a> (2005) <i>Postmodern Analysis, Third Edition,</i> Springer.  See Theorem 12.1 on page 157 for the monotone increasing case.</li>
<li><a href="/facts/Walter_Rudin/9mxRAECY">Rudin, Walter R.</a> (1976) <i>Principles of Mathematical Analysis, Third Edition,</i> McGraw–Hill.  See Theorem 7.13 on page 150 for the monotone decreasing case.</li>
<li>Thomson, Brian S.; Bruckner, Judith B.; <a href="/facts/Andrew_M._Bruckner/KmxSy8qF">Bruckner, Andrew M.</a> (2008) [2001]. <i>Elementary Real Analysis</i>. ClassicalRealAnalysis.com. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-1-4348-4367-8.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>Edwards 1994, p. 165. Friedman 2007, p. 199. Graves 2009, p. 121. Thomson, Bruckner & Bruckner 2008, p. 385. - Edwards, Charles Henry (1994) [1973]. Advanced Calculus of Several Variables. Mineola, New York: Dover Publications. ISBN 978-0-486-68336-2. <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>According to Edwards 1994, p. 165, "[This theorem] is called Dini's theorem because Ulisse Dini (1845–1918) presented the original version of it in his book on the theory of functions of a real variable, published in Pisa in 1878". - Edwards, Charles Henry (1994) [1973]. Advanced Calculus of Several Variables. Mineola, New York: Dover Publications. ISBN 978-0-486-68336-2. <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
</ol>

Dini's theorem open-in-new

Dini's theorem