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  "body": "\n## Proof\n\nWe bound the [[harmonic numbers|0D2Y6G4C4S]] at powers of $2$ from below, by induction.\n\n1. For $N \\ge 1$, $H_N = \\sum_{n=1}^{N} 1/n$. Since $H_{N+1} = H_N + 1/(N+1) \u003e H_N$, the sequence $(H_N)$ is increasing.\n2. Let $j \\ge 1$. The block of terms $1/n$ with $2^{j-1} \u003c n \\le 2^j$ has $2^j - 2^{j-1} = 2^{j-1}$ terms.\n3. Each term in that block satisfies $1/n \\ge 1/2^j$. So the block has sum at least $2^{j-1}/2^j = 1/2$.\n4. We claim $H_{2^k} \\ge 1 + k/2$ for every natural number $k$. For $k = 0$, $H_1 = 1$.\n5. Assume the claim for $k$. Then $H_{2^{k+1}} = H_{2^k} + \\sum_{2^k \u003c n \\le 2^{k+1}} 1/n$. By step 3 with $j = k + 1$, this is at least $1 + k/2 + 1/2$.\n6. By induction, steps 4 and 5 prove the claim for every $k$.\n7. Let $M$ be a real number. By the [[Archimedean property|0A74SX55QP]] there is a natural number $k$ with $2(M - 1) \u003c k$, that is, $1 + k/2 \u003e M$.\n8. Let $N_0 = 2^k$. For $N \\ge N_0$, steps 1, 4 and 7 give $H_N \\ge H_{N_0} \\ge 1 + k/2 \u003e M$.\n9. So $H_N \\to +\\infty$, in the sense of a [[sequence tending to infinity|09AZCV6GMC]].\n10. In particular $(H_N)$ is not bounded. [[Convergent sequences are bounded|0S4JF619V7]], so $(H_N)$ does not converge.\n11. The $H_N$ are the partial sums of the [[harmonic series|0CXTMAY5PX]]. So the harmonic series is a [[divergent series|0NBRP6093V]].\n\n## Idea\n\nGroup the terms in blocks that end at powers of $2$: $\\frac12$, then $\\frac13 + \\frac14$, then $\\frac15 + \\cdots + \\frac18$, and so on. Each block has sum at least $\\frac12$. Infinitely many halves add up to infinity.\n",
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