Editing Limit of a function (section)

=== In terms of sequences ===
For functions on the real line, one way to define the limit of a function is in terms of the limit of sequences.  (This definition is usually attributed to [[Eduard Heine]].) In this setting:
<math display=block>\lim_{x\to a}f(x)=L</math>
if, and only if, for all sequences {{mvar|x{{sub|n}}}} (with, for all {{mvar|n}}, {{mvar|x{{sub|n}}}} not equal to {{mvar|a}}) converging to {{mvar|a}} the sequence {{math|''f''(''x{{sub|n}}'')}} converges to {{mvar|L}}.  It was shown by [[Sierpiński]] in 1916 that proving the equivalence of this definition and the definition above, requires and is equivalent to a weak form of the [[axiom of choice]].  Note that defining what it means for a sequence {{mvar|x{{sub|n}}}} to converge to {{mvar|a}} requires the [[epsilon, delta method]].

Similarly as it was the case of Weierstrass's definition, a more general Heine definition applies to functions defined on [[subset]]s of the real line. Let {{mvar|f}} be a real-valued function with the domain {{math|''Dm''(''f'' )}}.  Let {{mvar|a}} be the limit of a sequence of elements of {{math|''Dm''(''f'' ) \ {''a''}.}}  Then the limit (in this sense) of {{mvar|f}} is {{mvar|L}} as {{mvar|x}} approaches {{mvar|a}}  
if for every sequence {{math|''x{{sub|n}}'' ∈ ''Dm''(''f'' ) \ {''a''} }} (so that for all {{mvar|n}}, {{mvar|x{{sub|n}}}} is not equal to {{mvar|a}}) that converges to {{mvar|a}}, the sequence {{math|''f''(''x{{sub|n}}'')}} converges to {{mvar|L}}.  This is the same as the definition of a sequential limit in the preceding section obtained by regarding the subset {{math|''Dm''(''f'' )}} of {{tmath|\R}} as a metric space with the induced metric.