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I have just finished some analysis which seems to rely on the property that if $ 0 < x \leq a$ then, as a tends to zero, x will tend to a. Is this a valid property? I was wondering if there might a formal proof of the property?

Thanks in advance

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    $\begingroup$ Does is matter that I have $\alpha < \beta \leq \gamma$ rather than $\alpha \leq \beta \leq \gamma$ ? This is still the same person, I'm having trouble commenting on answers for some reason... $\endgroup$
    – user110436
    Commented Nov 21, 2013 at 12:19
  • $\begingroup$ Have you tried to prove it using, say, $\epsilon$-$\delta$? $\endgroup$ Commented Nov 21, 2013 at 12:20
  • $\begingroup$ It doesn't. When taking limits, $<$ becomes $\leq$ as the example $\{1/n\}_n$ shows. $\endgroup$
    – Siminore
    Commented Nov 21, 2013 at 12:55
  • $\begingroup$ Your account got "split", this is due to your account being "unregistered", in which case the website tries to remember who you are based on some cookies/ip-address/etc magic which is very, very fragile and fails easily. Please register your account. Afterward you should visit this page to request your two user profiles be merged. Be sure to include a link to both this profile and this profile. $\endgroup$ Commented Nov 21, 2013 at 13:36

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It is the comparison theorem: if $\alpha \leq \beta \leq \gamma$ and $\lim \alpha = \lim \gamma=\ell$, then also $\lim \beta =\ell$.

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