Algebra of limits
If and , then and ; if moreover for every , then .
Lemma 1.8
Let , . Then and . If for every , then .
Product rule
Splitting reduces the claim to the two component convergences.
Since and , for a given there are and with for all and for all . The sequence is bounded, so for some , giving
for all . As the threshold can be rescaled, . The sum and reciprocal parts are exercises in the same style.
Function limits
The same rules hold for limits of functions. If have and at an accumulation point of , then
the quotient requiring for all and .
Related
Stated in
- Lemma 1.8§1.1 Basics