- #1
twoflower
- 368
- 0
Hi all,
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 the uniqueness of limit of sequence here, when [itex]f\left( x_{n} \right)[/itex] is not a sequence, but a function?
Thank you for the explanation.
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 the uniqueness of limit of sequence here, when [itex]f\left( x_{n} \right)[/itex] is not a sequence, but a function?
Thank you for the explanation.