Is it sufficient to prove $P(x) \ge (\text{or} \le) a$ if we already know $P(x) > (\text{or} <)a$?
For example, to prove $$ \forall n \ge 1 , \sum_{i=1}^{n}\frac{1}{i^2} \le 2 $$
Suppose I have already proved $$ \forall n \ge 1 , \sum_{i=1}^{n}\frac{1}{i^2} < \frac{7}{4} - \frac{1}{n} $$ Then, are we done? (Because $\frac{7}{4}-\frac{1}{n} < 2 \le 2$)
Of course. If "greater than" is true, then "greater than or equal to" is true. Look at the truth table under OR here: https://medium.com/i-math/intro-to-truth-tables-boolean-algebra-73b331dd9b94.