Let $\{p_n\}$ denote a sequence such that: $$ S_n = {1\over p_1} + {1\over p_2} + \cdots + {1\over p_n} $$ converges. Prove that: $$ \sigma_n = \left(1+{1\over p_1}\right)\left(1+{1\over p_2}\right)\cdots\left(1+{1\over p_n}\right) $$ converges, where $n, p_n \in \Bbb N$.
Consider each bracket from $\sigma_n$. By ${1\over p_n} > 0$: $$ \forall k \in \Bbb N: \left(1+{1\over p_k}\right) > 1 $$ So $\sigma_n$ must be monotonically increasing by: $$ {\sigma_{n+1} \over \sigma_n} = \left(1+{1\over p_{n+1}}\right) > 1 $$
To show a monotonic sequence is convergent it's sufficient to show that it's bounded above. Lets try to find the bound. Recall: $$ \ln(1+x) \le x \iff (1+x) \le e^x \tag1 $$ So by $(1)$ we have: $$ \sigma_n \le e^{S_n} $$ But $S_n$ is convergent! And thus: $$ \sigma_n \le e^L $$ where: $$ L = \lim_{n\to\infty}S_n $$ Which by monotone convergence theorem proves $\sigma_n$ is convergent.
I'm kindly asking to verify my proof and point to the mistakes in case of any. Thank you!