Question

Let (x_{n}) and(y_{n}) be sequences of real numbers.

(a) If (y_{n}) is bounded below and if x_{n} -> [infinity] , then (x_{y} + y_{n}) -> [infinity] Similarlyif (y_{n}) is bounded above and x_{n} -> - [infinity] , then (x_{n} + y_{n}) -> - [infinity]

Answer 1

The behavior of sequences approaching infinity or negative infinity is discussed, with the conclusion that the sum of a sequence tending to infinity and one that is bounded below also tends toward infinity, and similarly for the sum of a sequence tending to negative infinity with one that is bounded above.

Step-by-step explanation:

The question deals with sequences of real numbers and their behavior as they approach infinity or negative infinity. Specifically, it states that if we have a sequence (yn) which is bounded below and another sequence (xn) that tends towards infinity, then the sum of these sequences (xn + yn) will also tend towards infinity. This situation occurs because while (yn) is restricted from going below a certain value, it does not restrict the growth of (xn), allowing their sum to grow without bound. Similarly, if (yn) is bounded above and (xn) tends to negative infinity, the sum (xn + yn) will tend towards negative infinity for the same reasoning.