Proving that $2^{n}n+1$ for all integers $n1.$

Hypothesis: $2^{n}n+1$

Base Case: for $n=1$:$2^1=2,1+1=2,22,$ This base case holds true.

Inductive step: Assuming the statement is true for $n=k$ :

Inductive hypothesis: $2^{k}k+1$

To show that the statement is still true for $k+1$:$2^{k+1}(k+1)+1$

Using inductive hypothesis: $2^{k+1}22^k$

Since it was assumed $2^{k}k+1$ :

$22^{k}2(k+1),,2^{k+1}2k+2$

Need to show that $2k+2k+2,,2kk$

This is true for all $k1,$ proving that $2^{k+1}(k+1)+1$

This is a weak induction because it is assumed the statement is true and then

prove it for $k+1.$

Do you agree or disagree with the statement? and why?