You've uploaded an image containing several mathematical problems. I'll help you with these questions step-by-step. Here's a brief breakdown of the problems:

1. Problem 1: Prove a set operation identity: $A \cap (B \setminus C) = (A \cap B) \setminus (A \cap C)$.
2. Problem 2: Prove that for an injective function $f: A \to B$, if $E \subseteq A$, then $f^{-1}(f(E)) = E$.
3. Problem 3: Prove the inequality $2n - 3 < 2^{n-2}$ for all $n \geq 5$.
4. Problem 4: Prove that if $a \in \mathbb{R}$ and $a \cdot a = a$, then $a = 0$ or $a = 1$.
5. Problem 5: Prove that for $0 \leq a < b$, $a^2 \leq ab < b^2$.
6. Problem 6: Prove that if $S$ is bounded above and $b > 0$, then for $bS := \{bs : s \in S \}$, $\inf(bS) = b \cdot \inf(S)$.
7. Problem 7: Prove using the definition of a limit that $\lim_{n \to \infty} \frac{2n^2}{3n^2 + 1} = \frac{2}{3}.$

Which of these problems would you like to focus on first, or would you like to solve them all sequentially?

Let me know if you want further explanation on any topic.