# Prove inequality about sequences

## Homework Statement

Let $$u_n, v_n$$ be a sequence of positive real number such that
$$u_{n+1}\leq (1+v_n)u_n$$ where $$\sum_{n=1}^{\infty}v_n$$
converges.

It is true or false if true please help me prove.
if $$\frac{u_{n+1}}{u_n}\leq M$$ then $$u_n\leq M$$
for some $$M$$ be a positive real number.

whether we can find $$M$$ is a positive real number such that
$$u_n\leq M$$ for all natural number n

## Answers and Replies

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