Proposition:   limx → a c = c.
Proof:

  1.Insert here In the case of   limx → a c = c   this means that for all   ε > 0   we have to find a   δ > 0   such that if   0 < |x-a| < δ   then   |c-c| < ε.

  2.Insert here This proves that   limx → a c = c.

  3.Insert here Recall that   limx → a f(x) = L   means that for all   ε > 0   we are able to find a   δ > 0   such that if   0 < |x-a| < δ   then   |f(x)-L| < ε.

  4.Insert here Therefore we can choose   δ   to be any positive real number, e.g.   δ = 1.

  5.Insert here So let   ε   be any positive real number. Then the condition   |c-c| < ε   is always true since we are assuming that   ε > 0.

  6.Insert here With this (or any other) choice of   δ, it is true that if   |x-a| < 1   then   |c-c| < ε   (since the conclusion is always true).

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