1. The basis step. This involve showing that a statement is true for some natural number
\[k\]
.This step is labelled
\[P(k)\]
.2. Assume
\[P(m)\]
is true for \[m \ge k\]
then prove \[P(m+1)\]
is true.A very simple example is: Prove
\[n^2 \gt n\]
for \[n \gt 1\]
.
1. The basis step. \[P(2)\]
is true since \[2^2 \gt 2\]
.
2. Assume \[P(m)\]
is true for \[m \gt 2\]
.\[P(m+1)\]
is the statement \[(m+1)^2 \gt (m+1)\]
.Expanding the brackets gives
\[m^2+2m+1=m+1\]
Use
\[P(m)\]
which states \[m^2 \gt m\]
to give\[m+2m+1=m+1\]
which simplifies to
\[3m+1=m+1\]
This is true for
\[m \gt 2\]