1. **State the problem:** Prove by induction that for all natural numbers $n$, the inequality $$\frac{1}{5^{n+1}} + \frac{1}{5^{n+2}} + \cdots + \frac{1}{2 \cdot 5^n} > \frac{1}{2}$$ holds. 2. **Base case ($n=1$):** Evaluate the left side: $$\frac{1}{5^{1+1}} + \frac{1}{2 \cdot 5^1} = \frac{1}{5^2} + \frac{1}{2 \cdot 5} = \frac{1}{25} + \frac{1}{10} = \frac{2}{50} + \frac{5}{50} = \frac{7}{50} = 0.14$$ Check if $0.14 > 0.5$? No, so the problem statement as given seems inconsistent or needs clarification. 3. **Re-examining the problem:** The sum indices and terms are ambiguous. Possibly the sum is from $k=n+1$ to $k=2n$ of $\frac{1}{5^k}$, or the last term is $\frac{1}{2 \cdot 5^n}$ as a single term. 4. **Assuming the sum is:** $$\sum_{k=n+1}^{2n} \frac{1}{5^k} > \frac{1}{2}$$ 5. **Base case ($n=1$):** $$\sum_{k=2}^{2} \frac{1}{5^k} = \frac{1}{5^2} = \frac{1}{25} = 0.04$$ Not greater than $\frac{1}{2}$, so the inequality does not hold. 6. **Conclusion:** The inequality as stated does not hold for $n=1$, so induction cannot proceed. If the problem statement is clarified or corrected, please provide the exact sum and inequality.