Let P(n); 2n > n
when n = 1, 21 > 1, Hence, P(1) is true.
Let P(m) is true for n = m, i.e.,
P(m) : 2m > m ……(i)
Now, we have to proved that the given statement is also true for n = m + 1
i.e., P(m + 1) is true
Multiplying by 2 on both sides of equation (i), we have
2 × 2m > 2m
⇒ 2m+1 > 2m = m + m > m + 1
Thus, P(m + 1) is true.
Hence, by the principle of mathematical induction,
P(n) is true for each natural number n ∈ N.
Hence Proved.