3.1. Sequences

Examples 3.1.12:

Show that the sequence sin(n) / n and cos(n) / n both converge to zero.

This might seem difficult because trig functions such as sin and cos are often tricky. However, using the Pinching theorem the proof will be very easy.

We know that | sin(x) | 1 for all x. Therefore

-1 sin(n) 1
for all n. But then we also know that:
-1/n sin(n)/n 1/n
The sequences {1/n} and -1/n both converge to zero so that the Pinching theorem applies and the term in the middle must also converge to zero.

To prove the statement involving the cos is similar and left as an exercise.

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