Sentence examples for using the elementary inequality from inspiring English sources

By using the elementary inequality for, we have (2.24).

Finally, using the elementary inequality for real numbers (3.19).

Using the elementary inequality, ( and ) in (2.24), we get (2.25).

Taking small enough and using the elementary inequality, we obtain (3.24).

Now, using the elementary inequality for real numbers (4.5). on (4.6).

Using the elementary inequality ( reals) and equality (2.11), we have (2.12).

We have used the elementary inequality ∑ i ≥ 0 1 2 i α ≤ c α. Notice that this shows that { ( f ) B ( x, ϱ i ) } is a Cauchy sequence and therefore the following limit exists: lim i → ∞ ( f ) B ( x, ϱ i ).

As to the first term, invoke (29) and use the elementary inequality 1 − cos x x 2 ≤ 1 2 to obtain | E [ e A ˜ n t 2 ( cos ( B ˜ n t 3 ) − 1 ) 1 T c ] | ≤ 1 2 t 6 e t 2 / 3 E [ B ˜ n 2 ].

There is another proof using the elementary amalgamation theorem and the elementary chain theorem.

Consequently, we can apply the elementary inequality (226).

Thus, using Lemma 2.4 and the elementary inequality, where is any constant, we have (2.11).