Prove: If lim n → ∞ a n = L and a n &gt; a for all n then L ≥ a

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Prove: If       lim          n      →      ∞            a    n    =  L and       a    n    &amp;gt;  a for all n then   L  ≥  a

eishale2n · Accepted Answer

Suppose that   L  &amp;lt;  a. We put   ε  =  a  −  L  &amp;gt;  0. Since             lim              n        →        ∞                    a      n        =    L  , there exists       N    0    ∈      N   such that      |        a    n    −  L      |    &amp;lt;  ε    ∀  n  ≥      N    0    .Then       a    n    −  L  &amp;lt;  a  −  L for all   n  ≥      N    0  , or       a    n    &amp;lt;  L. This is contradict to the assumption       a    n    &amp;gt;  a for all n. Hence   L  ≥  a. In the above argument, we have seen that we only need       a    n    &amp;gt;  L for sufficiently large n.Moreover, in the general case we do not have   L  &amp;gt;  a. Indeed, we observe that although             a      n        =          1      n        &amp;gt;    0   for all n but       L    =          lim              n        →        ∞                    a      n        =          lim              n        →        ∞                    1      n        =    0

Prove: If lim_(n => infty)a_n=L and a_n>a for all n then L >= a

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