1. 上确界也称为最小上界。对于![](https://cdn.nlark.com/yuque/__latex/e1e1d3d40573127e9ee0480caf1283d6.svg#card=math&code=R&id=kLD2k)的非空子集![](https://cdn.nlark.com/yuque/__latex/5dbc98dcc983a70728bd082d1a47546e.svg#card=math&code=S&id=IucQD),如果该集合中的任意元素![](https://cdn.nlark.com/yuque/__latex/03c7c0ace395d80182db07ae2c30f034.svg#card=math&code=s&id=uUUVo)均有 ![](https://cdn.nlark.com/yuque/__latex/a3a13ccd969c10b7574878195345b4fd.svg#card=math&code=s%20%5Cleq%20t&id=ofu3s)，，即![](https://cdn.nlark.com/yuque/__latex/e358efa489f58062f10dd7316b65649e.svg#card=math&code=t&id=I5TFs)是集合的一个上界，且![](https://cdn.nlark.com/yuque/__latex/e358efa489f58062f10dd7316b65649e.svg#card=math&code=t&id=YJSEH)是满足此不等式条件的最小值，即![](https://cdn.nlark.com/yuque/__latex/e358efa489f58062f10dd7316b65649e.svg#card=math&code=t&id=l6R9C)为集合![](https://cdn.nlark.com/yuque/__latex/5dbc98dcc983a70728bd082d1a47546e.svg#card=math&code=S&id=rzhO1)的上确界。如果满足此条件的![](https://cdn.nlark.com/yuque/__latex/e358efa489f58062f10dd7316b65649e.svg#card=math&code=t&id=hTNNn)不存在，则称此集合的上确界为![](https://cdn.nlark.com/yuque/__latex/9ab0347369b93587a1fc8dbd6c6a8862.svg#card=math&code=%2B%5Cinfty&id=kGjq6)。如果上确界存在，则必定唯一<br />        下确界也称为最打小界。![](https://cdn.nlark.com/yuque/__latex/e358efa489f58062f10dd7316b65649e.svg#card=math&code=t&id=PO8qp)是集合![](https://cdn.nlark.com/yuque/__latex/5dbc98dcc983a70728bd082d1a47546e.svg#card=math&code=S&id=Z8NJ4)的一个下界，即对于![](https://cdn.nlark.com/yuque/__latex/5dbc98dcc983a70728bd082d1a47546e.svg#card=math&code=S&id=GNIyu)中的任意元素![](https://cdn.nlark.com/yuque/__latex/03c7c0ace395d80182db07ae2c30f034.svg#card=math&code=s&id=HsQ2z)均有![](https://cdn.nlark.com/yuque/__latex/5a9847d45ba8689048f6b2e1baa567a1.svg#card=math&code=s%5Cgeq%20t&id=APjOE),且![](https://cdn.nlark.com/yuque/__latex/e358efa489f58062f10dd7316b65649e.svg#card=math&code=t&id=KxaEW)是最大下界，则称![](https://cdn.nlark.com/yuque/__latex/e358efa489f58062f10dd7316b65649e.svg#card=math&code=t&id=PSmOU)是![](https://cdn.nlark.com/yuque/__latex/5dbc98dcc983a70728bd082d1a47546e.svg#card=math&code=S&id=Ksn7q)最大下确界