Let be a sequence of positive real numbers such that

(1)

Prove that converges.

**Solution**

Fix and let . Then, there exist such that where . Thus,

Letting it follows that

Since this holds forall it follows that and the result follows.

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Let be a sequence of positive real numbers such that

(1)

Prove that converges.

**Solution**

Fix and let . Then, there exist such that where . Thus,

Letting it follows that

Since this holds forall it follows that and the result follows.