For P(k+1)P( k+1 ), the LHS =k+1∑i=11i×(i+1)=k∑i=11i×(i+1)+1(k+1)×(k+2)=k(k+1)+1(k+1)×(k+2)=\displaystyle \sum_{i=1}^{k+1} \frac{1}{i\times(i+1)}= \sum_{i=1}^{k} \frac{1}{i\times(i+1)} + \frac{1}{(k+1)\times(k+2)} =  \frac{k}{(k+1)} + \frac{1}{(k+1)\times(k+2)} .

Using the direct proof technique, we assume that P(k)≡TP( k ) \equiv T so k∑i=11i×(i+1)=k(k+1)\displaystyle \sum_{i=1}^k \frac{1}{i\times(i+1)}= \frac{k}{(k+1)}.

Define predicate P(n)P( n ) as the equation n∑i=11i×(i+1)=nn+1\displaystyle \sum_{i=1}^n \frac{1}{i\times(i+1)}= \frac{n}{n+1}, where the domain of nn is Z+Z^+ and the smallest element in the domain is 1.

Basis step: to prove that P(1)≡TP( 1 ) \equiv T.

proof

Inductive step: to prove that ∀k(P(k)→P(k+1))≡T\forall k ( P( k ) \to P( k+1 ) ) \equiv T.

To use the UG rule, we consider an arbitrary positive integer kk.

For P(1)P( 1 ), the left-hand side (LHS) =1∑i=11i×(i+1)=12==\displaystyle \sum_{i=1}^1 \frac{1}{i\times(i+1)}= \frac{1}{2}= the right-hand side (RHS), so P(1)≡TP( 1 ) \equiv T.

Since k(k+1)+1(k+1)×(k+2)=k×(k+2)+1(k+1)×(k+2)=(k+1)2(k+1)×(k+2)=(k+1)(k+2)\displaystyle \frac{k}{(k+1)} + \frac{1}{(k+1)\times(k+2)} =  \frac{k\times(k+2)+1}{(k+1)\times(k+2)} = \frac{(k+1)^2}{(k+1)\times(k+2)} = \frac{(k+1)}{(k+2)}, P(k+1)≡TP( k+1 ) \equiv T.

QED

Inductive step: to prove that ∀k((P(a)∧⋯∧P(k))→P(k+1))≡T\forall k ( (P(a) \wedge \cdots \wedge P( k )) \to P( k+1 ) ) \equiv T, where k≥bk \geq b.

Basis step: to prove that (P(a)∧⋯∧P(b))≡T(P( a ) \wedge \cdots \wedge P(b)) \equiv T, where aa is the smallest element in the domain of nn and b≥ab\geq a is an element in the domain of nn.

Define predicate P(n)P( n ) with the domain of nn and the smallest element aa in the domain.

proof

QED

Basis step: to prove that P(a)≡TP( a ) \equiv T, where aa is the smallest element in the domain of nn.

Define predicate P(n)P(n)P( n ) with the domain of nn and the smallest element aa in the domain.

Inductive step: to prove that ∀k(P(k)→P(k+1))≡T∀k(P(k)→P(k+1))≡T\forall k ( P( k ) \to P( k+1 ) ) \equiv T.

proof

QED