Example 2

Tag 0BTD, permanent. Cite it as [Bourbaki, Tag 0BTD].

Let $(a_n)$ be a summable sequence of positive numbers. Since

\lim_{n \to \infty} a_n = 0,

there exists an integer

n_0

such that

a_n \leq 1

whenever

n \geq n_0

. Consequently, for each

n \geq n_0

we have

a_n^2 \leq a_n

, and therefore the sequence

(a_n^2)

is summable in

\mathbf{R}

. So is the sequence

(a_n^p)

for each integer

p > 1

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

Cited by

Nothing.

Nothing.