Limit Properties¶
As before, “sequences” here implicitly mean “in an ordered field”. Where a complete ordered field is assume, it will be explicitly noted.
Assume that \(a_n\) and \(b_n\) are sequences which converge to \(a\) and \(b\) respectively.
and so \(|a_n-a|=|(-a_n)-(-a)|\), so \(-a_n\) converges to \(a\).
If we choose \(N\) such that the appropriate tails of \(a_n\) and \(b_n\) are :math:epsilon/2:-close to \(a\) and \(b\) respectively, then the sum above will be \(<\epsilon/2+\epsilon/2=\epsilon\). Therefore, \(a_n+b_n\) converges to \(a+b\) (or in other words, the limit of a sum is the sum of the limits). This technique is known as the “half-\(\epsilon\), although in many applications, dividing by more than two is necessary.
What about multiplication?
Note that since \(a_n\) converges, it is bounded from both above and below, and so \(|a_n|\) is bounded by a number, \(M\).
Define \(N\) such that the \(N\)-tail has \(|b_n-b|<\epsilon/2(M+1)\) and \(|a_n-a|`<\epsilon/2(|b|+1)\) (we add \(1\) to avoid dividing by zero). Then for \(n>N\)
So the limit of a product is the product of the limits.
If \(a_n\) converges to \(a>0\), then a tail will be bounded from below by \(a-a/2=a/2\). Since the following only deals with tail properties, a simplifying assumption is that \(a_n>a/2\) for all \(n\). In particular, \(a_n\ne 0\) and so \(a_n^{-1}\) is well defined.
If we take \(N\) for \(a_n\) to be \(\epsilon(2|a|^2)\)-close to \(a\), then the expression above is \(<\epsilon\).
If \(a<0\), then \(-a_n\) converges to \(-a>0\), \(1/-a_n=-1/a_n\) converges to \(1/-a=-1/a' and so :math:`1/a_n\) converges to \(1/a\). In summary, the limit of the inverse is the inverse of the limit.
If \(b>a\), let \(\epsilon=(b-a)/2\) and find \(N\) where for \(n>N\) both \(a_n\) is \(\epsilon\)-close to \(a\) and \(b_n\) is \(\epsilon\)-close to \(b\), then \(b_n-a_n>b-\epsilon-(a+\epsilon)=b-a-2\epsilon=d-d=0\), so \(b_n>a_n\) at the tail. In particular, if \(a_n\geq b_n\) on a tail, then it cannot be the case that \(b>a\) so \(a\geq b\).
The summary of these results is that in an ordered field, limits preserve the ordered field structure (weakly, for the order).
Assume \(a_n\) and \(c_n\) both converge to \(a\), and \(a_n\leq b_n\leq c_n\). For a tail, \(a-\epsilon<a_n\leq b_n \leq c_n<a+\epsilon\), so \(b_n\) also converges to \(a\). This result is called “the sandwich theorem”, and is often a powerful aid to finding and proving limits.