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Suppose $a\le x_n\le b$ for all $n$ and suppose further that $x_n\rightarrow L$. Prove: $L \in [a, b]$.

The book provides a hint: if $L\lt a$ or if $L \gt b$, obtain a contradiction.

My reasoning is if $x_n\rightarrow L$, then $\left|x_n - L \right|\lt\epsilon$. (Definition of a limit). The question says $x_n\in [a, b]$.

So I put the sequence between the bounds $a$ and $b$.

$a\le\left|x_n - L \right|\le b$. When I break apart the absolute value brackets, this is the result:

$a- L \lt x_n\lt b+L$.

But this can't be true considering $x_n\in[a, b]$. Here, the left side is $a-L$ (where $L \lt a$) is too small to be between $[a, b]$. Same for the right side, $b+L$ (where $L \gt b$) is too large to be between $[a,b]$.

I'm not sure if I'm on the right track or not. Can someone help lead me to a better answer.

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    $\begingroup$ Where did you get $a \le |x_n−L| \le b$? That is wrong. $\endgroup$ – NickD May 2 '17 at 17:09
  • $\begingroup$ I'm assuming that. What do you suggest? $\endgroup$ – ErinA May 2 '17 at 17:13
  • $\begingroup$ You cannot assume something that is not true: what you know is that $a \le x_n \le b$. As the answer points out that means that $a - L \le x_n - L \le b - L$ and that blatantly contradicts your assumption. $\endgroup$ – NickD May 2 '17 at 17:16
  • $\begingroup$ What you can assume in order to argue by contradiction is that L is not in the interval $[a, b]$ and then show that that leads to a contradiction. If L is not in that interval, what does that tell you about $L$? Hint: there are two cases - in each case you should be able to say something about either $a -L$ or $b - L$ that contradicts one of the inequalities if n is large enough. $\endgroup$ – NickD May 2 '17 at 17:23
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Suppose $L<a$. Set $\varepsilon=(a-L)/2$. By definition of limit, there exists $N$ such that, for $n>N$, $$ |x_n-L|<\varepsilon $$ which implies $$ x_n-L<\frac{a-L}{2} $$ so $$ x_n<L+\frac{a-L}{2}=\frac{a+L}{2}<\frac{a+a}{2}=a $$ This is a contradiction.

Can you do the case $L>b$?

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  • $\begingroup$ This is what my teacher's assistant said to consider! But I had no idea how to work it in. Thank you so much for this! $\endgroup$ – ErinA May 2 '17 at 18:08
  • $\begingroup$ Is the contradiction $x_n$ $\lt$ $\frac{a+L} {2}$ $\lt$ a when it should be $x_n$ $\lt$ $\frac{a+L} {2}$ $\lt$ $\epsilon$ ? $\endgroup$ – ErinA May 2 '17 at 20:11
  • $\begingroup$ @ErinA No: the contradiction is in the fact that, by assumption, $x_n\ge a$. $\endgroup$ – egreg May 2 '17 at 20:32
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Your line $a \leq x_n - L \leq b$ should be $a-L \leq x_n - L \leq b-L$

Since $a \leq x_n \leq b$, it follows that $a - L \leq x_n - L \leq b- L $.

See what you can do with this now, and try to show that when $x$ is arbitrarily close to $L$, there are $x_n$ that leave [a,b], which is a contradiction.

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