Answer
See below.
Work Step by Step
Use proof by contradiction: suppose not, that is "there are no three successive integers whose sum is at least 45."
Divide the numbers into groups of threes, there will be a total of 10 groups with a total sum $S\lt 10(45)=450$
As we know the sequence is from 1 to 30, we can calculate the sum using a formula: $S(30)=\frac{30}{2}(1+30)=465\gt450$
Thus we have a contradiction and we can conclude that the original statement is true.