Let be a real sequence such that

If is a real sequence that is square summable; i.e the sequence converges.

Prove that is also square summable.

**Solution**

Let be defined as

where . We note that

Equality holds when . Hence, . From the hypothesis, it follows that is pointwise bounded. It follows from the Uniform boundedness principle ( Banach – Steinhaus ) that are bounded. Hence, is square summable.