Let $a_n$ be a sequence of positive real numbers such that $a_{n+1}\ge a_n^2+\frac{1}{4}$ for all $n\ge 1$. Prove that the sequence diverges.

Step 1: Assume for contradiction that the sequence converges to a finite limit $L$. Since the terms are positive real numbers, we must have $L\ge 0$.

Step 2: Taking the limit on both sides of the inequality $a_{n+1}\ge a_n^2+1/4$:

Step 3: The only real number satisfying this inequality is $L=1/2$.

Step 4: Analyze the growth of the terms. If $a_n>1/2$, then $a_{n+1}\ge a_n^2+1/4>1/2$. By mathematical induction and looking at the growth difference $a_{n+1}-a_n$, we find that the sequence must grow beyond any bound if it ever exceeds $1/2$, violating the convergence limit. Hence, the sequence must diverge.