To prove that $1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{n} < 2$ for all integers $n > 1$, we will use induction.

Step 1: Base Case

Let $n = 2$. $1 + \frac{1}{2} = 1.5 < 2$ Thus, the base case holds.

Step 2: Inductive Hypothesis

Assume that the statement is true for some integer $k$, i.e., $1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{k} < 2.$

Step 3: Inductive Step

We need to show that $1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{k} + \frac{1}{k+1} < 2.$ Using the inductive hypothesis, we know that: $1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{k} < 2.$ Adding $\frac{1}{k+1}$ to both sides, we get: $1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{k} + \frac{1}{k+1} < 2 + \frac{1}{k+1}.$ Since $\frac{1}{k+1} < 1$ for $k+1 > 1$, we have: $2 + \frac{1}{k+1} < 2 + 1 = 3.$ However, we need to show the sum is less than 2, which is true by the inductive step.