Answer
$n\leq F_n$ when $n\geq5$, see explanations.
Work Step by Step
Step 1. Find the relation between $n$ and $F_n$ of the Fibonacci sequence: $n=1,2,3,4,5,6,...$, $F_n=1,1,2,3,5,8$,
we can see that $n\leq F_n$ when $n\geq5$
Step 2. Recall the property of Fibonacci sequence, we have: $F_{n}=F_{n-1}+F_{n-2}$ and $F_1=F_2=1$
Step 3. Prove the statement is true when $n=5$: $5=F_{5}$, thus it is true for $n=5$.
Step 4. Assume the statement is true when $n=k$ ($k\gt5$): we have $k\leq F_{k}$
Step 5. Prove it is true for $n=k+1$: $RHS=F_(k+1)=F_k+F_{k-1}\geq k+F_{k-1}$, as $F_{k-1}\geq1$, we have $RHS\geq k+1=LHS$
Thus, the statement is also true for $n=k+1$
Step 6. Conclusion: with mathematical induction, we have proved that the statement is true for all natural numbers $n$.
