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Prove: If $\lim_{n\to\infty}a_n = L$ and $a_n > a$ for all $n$ then $L \geq a$

Proof: We know from the definition of the limit that $\forall_{\epsilon > 0} \exists_n s.t. \forall_{n>N} |a_n - L| < \epsilon$. Now since $a_n > a$ for all $n$...

I am not really sure where to go from here. Is it the case that all sequences defined by this statement are monotone non-increasing? Then intuitively we could say $a_n = L$ for sufficiently large $n$. Thus, by transitivity $L > a$

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Suppose that $L. We put $\varepsilon=a-L>0$. Since $\displaystyle\lim_{n\rightarrow\infty}a_n=L$, there exists $N_0\in \mathbb{N}$ such that $ |a_n-L|<\varepsilon \quad\forall n\geq N_0. $ Then $a_n-L for all $n\geq N_0$, or $a_n for all $n\geq N_0$. This is contradict to the assumption $a_n>a$ for all $n$. Hence $L\geq a$. In the above argument, we have seen that we only need $a_n>L$ for sufficiently large $n$.

Moreover, in the general case we do not have $L>a$. Indeed, we observe that although $\displaystyle a_n=\frac{1}{n}>0$ for all $n$ but $\displaystyle L=\lim_{n\rightarrow\infty}a_n=\lim_{n\rightarrow\infty}\frac{1}{n}=0$.

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    Where and what is the extended discussion?2012-10-05
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Hint: To show that $L\geqslant a$, it is enough to show that $a\leqslant L+\varepsilon$, for every $\varepsilon\gt0$. Now, $a_n\to L$ hence, for every $\varepsilon\gt0$, there exists $n_\varepsilon$ such that for every $n\geqslant n_\varepsilon$, $a_n$ is such that...

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    For any set $S$, [$a\leqslant s$ for every $s$ in $S$] is logically equivalent to [$a\leqslant\inf S$]. Is this your question?2012-10-01
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The sequence need not be monotone. But, here's a hint for one approach towards a proof:

Try arguing by contradiction: Assume $L. Now choose an $\epsilon>0$ so that $L+\epsilon. What can you say about the terms $a_n$ of the sequence for large $n$?

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    @CodeKingPlusPlus For large $n$, you would know |a_n-L|<\epsilon; and this would imply a_n. But L+\epsilon, so...2012-09-29