Original Post.

**Titu’s Lemma**

It is just a variant of Cauchy Schwartz inequality. It states that for and we will have:

**Generalisation**

We have for and :

*Hint for proof*

Just use Cauchy

**Applications**

Using Titu’s lemma one can solve a variety of problems.A list of such problems:

1.

Prove that for positive reals we have:

(NESSBITT)

2.

For positive unequal reals we have:

3.

Prove that

for all reals

4.

are positive numbers such that Prove that:

5.

For positive reals we have:

6.

Prove that

7.

Prove that:

8.

Let for . Prove that:

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