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I'm trying to prove that if $\{ a_n \}$ is a monotonically decreasing sequence, then $\lim(a_n) = \inf\{ a_n \ | \ n \in \mathbb{N} \}$. Here's my proof:

Suppose that $\lim(a_n) \neq \inf\{ a_n \ | \ n \in \mathbb{N} \}$. Let's say that $\lim(a_n) = L$. Since $\{ a_n \}$ is monotonically decreasing, it must be that $L < \inf\{ a_n \ | \ n \in \mathbb{N} > \}$. Thus, we have two cases to examine:

Case 1: $L \in \mathbb{R}$. Let $d = \inf\{ a_n \ | \ n \in \mathbb{N} > \} - L$ and let $\epsilon = d/2$. Since $(a_n) \to L$, there exists an $N$ such that $n>N$ implies that $|a_n-L| < \epsilon$. Thus, there exists an element $a_m$ (where $m>N$) such that $a_m < \inf\{ a_n \ | > \ n \in \mathbb{N} \}$. This contradicts the definition of $\inf$ and so it must be that $\lim(a_n) = \inf\{ a_n \ | \ n \in \mathbb{N} \}$.

Case 2: $L = -\infty$. If $\lim(a_n) = -\infty$, then for any $M$, there exists an $N$ such that $n>N$ implies that $a_n < M$. Letting $M > = \inf\{ a_n \ | \ n \in \mathbb{N} \}$, we see that there exists an element $a_m$ (where $m>N$) such that $a_m < \inf\{ a_n \ | \ n \in > \mathbb{N} \}$. Again, this contradicts the definition of $\inf$ and so it must be that $\lim(a_n) = \inf\{ a_n \ | \ n \in \mathbb{N} \}$.

Would anyone mind verifying that this is correct?

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  • $\begingroup$ The case where $L = \infty$ is fine, but I think more effort is needed when $L \in \mathbb{R}$. First, if you are using your method, you should prove that $d > 0$. Also, after choosing an element $a_m$ such that $a_m < \inf_{n \in \mathbb{N}} a_n$, you should add that this is a contradiction because $(a_n)_{n \in \mathbb{N}}$ is monotonically decreasing since this result may not be true otherwise. $\endgroup$ – user43378 May 14 '14 at 1:57
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My biggest problem with the proof is that you take for granted the existence of the limit. Before differentiating between the cases for $L$ you must first show that $\lim (a_n)$ exists. We don't still know if the sequence converges. And even then I think a lot more needs to be done to show why $L \gt \inf \{a_n\}$ is not plausible - it is not obvious to me. This is how I would get around it.

Start with what is given. All we know is that the sequence is monotone decreasing and that the infimum exists. So let us use that. Let $\epsilon \gt 0$ be arbitrary. Then $ \inf \{a_n\} + \epsilon \gt \inf \{a_n\}$. This means there is an element $a_m \in \{a_n\}$ such that $ a_m \lt \inf \{a_n\} + \epsilon $ or else $\inf \{a_n\} + \epsilon$ will be the infimum of $\{a_n\}$ leading to a contradiction. Now,

$n \ge m \implies \inf \{a_n\} - \epsilon \lt \inf \{a_n\} \le a_n \le a_m \lt \inf \{a_n\} + \epsilon \implies |a_n - \inf \{a_n\}| \lt \epsilon$

$\mathscr{Q.E.D.}$

Now we have avoided assuming the existence of the limit. Instead we have proved that the infimum is in fact the limit and hence it exists. Hope I helped.

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