Verify Bounded Convergence Theorem for the sequence of functions

Question

Verify Bounded Convergence Theorem for the sequence of functions $\left \{ f_n \right \}$ where

$f_n (x) = \frac{1}{\left ( 1 + \frac{x}{n} \right )^n } , 0\leq x \leq 1, n \in N$

26 Feb 2024

Answer :

Word Count : 262

To verify the Bounded Convergence Theorem for the sequence of functions $\{ f_n \}$, we need to check two conditions:

1. Pointwise convergence: Show that the sequence of functions converges pointwise to a limit function $f(x)$.
2. Uniform boundedness: Show that the sequence of functions is uniformly bounded.

Let's proceed with each step:

1. **Pointwise convergence:**
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