Prove $1+2+3+...+n<=n^2$ for all n∈N 
Prove $1+2+3+...+n<=n^2$ for all n∈N

This is what I have so far. Not sure what to do now.

One potential stumbling block is that your "in other words" isn't quite complete.  What "$P(k)$ implies $P(k+1)$" really says is that if 
$$
1+2+\cdots+k \leq k^2
$$
then
$$
1+2+\cdots+k+(k+1) \leq (k+1)^2
$$
In other words (!), you use the premise that $1+2+\cdots+k \leq k^2$ as the starting point.  If you add $k+1$ to both sides, you get
$$
1+2+\cdots+k+(k+1) \leq k^2+(k+1)
$$
Can you finish the line of reasoning?
