Question:
$\text{Prove by induction that, for all integers } n, n \geq 1:$ $\sum\limits_{r=1}^{n} r >\frac{1}{2}n^2$
Working:
Step 1 (Prove true for n=1): $1>\frac{1}{2}(1)^2$
Step 2 (Assume true for n=k): $ k >\frac{1}{2}k^2$
Step 3 (Prove true for n=k+1):
And having only faced equations with an equals (=) sign I have no idea what to do next. Right now I have assumed that it stands true for $k$ and I will try to prove for $k+1$. What should be my next step?