1 1 ! + 1 2 ! + 1 3 ! + ⋯ + 1 n ! ≤ 2 − 1 n {\displaystyle {\frac {1}{1!}}+{\frac {1}{2!}}+{\frac {1}{3!}}+\cdots +{\frac {1}{n!}}\leq 2-{\frac {1}{n}}}
L : 1 n ! = 1 1 = 1 R : 2 − 1 n = 2 − 1 = 1 1 = 1 {\displaystyle {\begin{aligned}L:{\frac {1}{n!}}={\frac {1}{1}}=1\\R:2-{\frac {1}{n}}=2-1=1\\1=1\\\end{aligned}}}
1 1 ! + 1 2 ! + 1 3 ! + ⋯ + 1 k ! ≤ 2 − 1 k {\displaystyle {\frac {1}{1!}}+{\frac {1}{2!}}+{\frac {1}{3!}}+\cdots +{\frac {1}{k!}}\leq 2-{\frac {1}{k}}}
1 1 ! + 1 2 ! + 1 3 ! + ⋯ + 1 k ! ⏟ = ( e n o u g h t o s a y ) 2 − 1 k + 1 ( k + 1 ) ! ≤ 2 − 1 k + 1 2 − 1 k + 1 ( k + 1 ) ! ≤ 2 − 1 k + 1 / + 1 k 1 ( k + 1 ) ! ≤ − 1 k + 1 + 1 k / ∗ ( k + 1 ) ! k ! k + 1 = 1 ∗ 2 ∗ 3 ∗ ⋯ ∗ ( k − 1 ) ∗ k ∗ ( k + 1 ) k + 1 = 1 ∗ 2 ∗ 3 ∗ ⋯ ∗ ( k − 1 ) ∗ k = k ! k ! k = 1 ∗ 2 ∗ 3 ∗ ⋯ ∗ ( k − 1 ) ⧸ k ( k + 1 ) ⧸ k = 1 ∗ 2 ∗ 3 ∗ ⋯ ∗ ( k − 1 ) ∗ ( k + 1 ) = ( k − 1 ) ! ( k + 1 ) ↓ 1 ≤ − k ! + ( k − 1 ) ! ( k + 1 ) / ( c h a n g e − p l a c e s ) 1 ≤ ( k − 1 ) ! ( k + 1 ) − k ! o p e n − b r a c k e t s : ( k − 1 ) ! ( k + 1 ) = ( k − 1 ) ! × k + ( k − 1 ) ! × 1 = k ( k − 1 ) ! + ( k − 1 ) ! ↓ 1 ≤ k ( k − 1 ) ! + ( k − 1 ) ! − k ! 1 ≤ k ! + ( k + 1 ) ! − k ! 1 ≤ ( k − 1 ) ! 1 ≤ 1 ∗ 2 ∗ 3 ∗ ⋯ ∗ ( k − 1 ) {\displaystyle {\begin{aligned}&\underbrace {{\frac {1}{1!}}+{\frac {1}{2!}}+{\frac {1}{3!}}+\cdots +{\frac {1}{k!}}} _{=(enoughtosay)2-{\frac {1}{k}}}+{\frac {1}{(k+1)!}}\leq 2-{\frac {1}{k+1}}\\&2-{\frac {1}{k}}+{\frac {1}{(k+1)!}}\leq 2-{\frac {1}{k+1}}/+{\frac {1}{k}}\\&{\frac {1}{(k+1)!}}\leq -{\frac {1}{k+1}}+{\frac {1}{k}}/*(k+1)!\\&\color {red}{\frac {k!}{k+1}}={\frac {1*2*3*\cdots *(k-1)*k*(k+1)}{k+1}}=1*2*3*\cdots *(k-1)*k=k!\\&\color {red}{\frac {k!}{k}}={\frac {1*2*3*\cdots *(k-1)\not {k}(k+1)}{\not {k}}}=1*2*3*\cdots *(k-1)*(k+1)=(k-1)!(k+1)\\&\downarrow \\&1\leq -k!+(k-1)!(k+1)/(change-places)\\&1\leq (k-1)!(k+1)-k!\\&open-brackets:(k-1)!(k+1)=(k-1)!\times k+(k-1)!\times 1=k(k-1)!+(k-1)!\\&\downarrow \\&1\leq k(k-1)!+(k-1)!-k!\\&1\leq k!+(k+1)!-k!\\&1\leq (k-1)!\\&1\leq 1*2*3*\cdots *(k-1)\\\end{aligned}}}
הביטוי 1 1 ! + 1 2 ! + 1 3 ! + ⋯ + 1 k ! ≤ 2 − 1 k {\displaystyle {\frac {1}{1!}}+{\frac {1}{2!}}+{\frac {1}{3!}}+\cdots +{\frac {1}{k!}}\leq 2-{\frac {1}{k}}} נכון עבור כל n {\displaystyle \ n} טבעי החל מ n = 1 {\displaystyle \ n=1} על פי 3 שלבי האינדוקציה.