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I don't fully understand the proof of uniqueness of limit of function. Our teacher proved it using the Heine's theorem. Here it is:

Proof:

Let [itex]\lim_{x \rightarrow a} f(x) = A[/itex] and [itex]\lim_{x \rightarrow a} f(x) = B[/itex].

Let [itex]\left{ x_{n} \right}[/itex] satisfies: [itex]\lim_{n \rightarrow \infty} x_{n} = a[/itex]. Then

[tex]

\Longrightarrow^{Heine}

\begin{array}{cc}\lim f\left( x_{n} \right) = A\\\lim f\left( x_{n} \right) = B\end{array}\right

[/tex]

Ok, I understand this, because according to Heine it's equivalent. But I don't get the next step:

[tex]

\Longrightarrow^{\mbox{Uniqueness of limit of sequence}} A = B \Longrightarrow^{Heine 2} \mbox{Uniqueness of limit of function}

[/tex]

How can I use theuniqueness of limit of sequencehere, when [itex]f\left( x_{n} \right)[/itex] is not a sequence, but a function?

Thank you for the explanation.

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# Homework Help: Please explain use of Heine in proof of simple theorem

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