How Do You Prove the Inequality Involving Primes and Harmonic Sum for POTW #337?

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Here is this week's POTW:

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Suppose $q_1,\ldots, q_r$ are the primes in the interval $[1, n]$ where $n$ is an integer $> 1$. Prove

$$\prod_{j = 1}^r \left(1 - \frac{1}{q_j}\right)\sum_{k = 1}^n \frac{1}{k} < 1$$-----

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No one answered this week's problem. You can read my solution below.

Spoiler

Note $$\prod_{j = 1}^r \frac{1}{1 - \frac{1}{q_j}} = \sum_{m_1 \ge 0} \sum_{m_2 \ge 0}\cdots \sum_{m_r \ge 0} \frac{1}{q_1^{m_1}q_2^{m_2} \cdots q_r^{m_r}} = \sum_{k\in S} \frac{1}{k}$$ where $S$ is the set of all positive integers $k$ whose prime factors are no greater than $n$. Since $$\sum_{k\in S} \frac{1}{k} > \sum_{k = 1}^n \frac{1}{k}$$it follows that $$\prod_{j = 1}^r \frac{1}{1 - \frac{1}{q_j}} > \sum_{k = 1}^n \frac{1}{k}$$ and therefore $$\prod_{j = 1}^r \left(1 - \frac{1}{q_j}\right) \sum_{k = 1}^r \frac{1}{k} < 1$$ as desired.