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Put $N=\lfloor x\rfloor$. Comparing the harmonic number with an integral gives $H_N={\sum }_{n=1}^{N}1/n>{\int }_1^{N+1}dt/t=\log (N+1)>\log x$. By the fundamental theorem of arithmetic, the finite Euler product ${\prod }_{p\le x}(1-p^{-1}{)}^{-1}$ expands as the sum of $1/n$ over positive integers whose prime factors are at most $x$. It therefore includes every term of $H_N$, andThe first strict inequality is the geometric-series bound for the logarithmic remainder; the last uses the telescoping series ${\sum }_{n=2}^{\infty }1/[n(n-1)]=1$. Thus the prime reciprocal lower bound is