Question

Explain why a sequence that is bounded above has infinitely many upper bounds.

Short answer

Any real number $k>M$which is the upper bound of the sequence.

Hence, if the sequence is bounded above, it has infinitely many upper bounds.

Step-by-step solution

Step 1. Given information

We need to explain why a sequence that is bounded above has infinitely many upper bounds.

Step 2. Consider the sequence ak that is bounded above.

So, according to definition there exists a real number $M$such that $M>a_k$for every $k$.

Therefore,$M$is the upper bound of the sequence.

Step 3. Now let there exists N, such that M<N then N>ak.

Therefore, $N$is also an upper bound of the sequence.

Hence, any real number $k>M$is the upper bound of the sequence.

Hence, if the sequence is bounded above, it has infinitely many upper bounds.