Question

Answer 1

Since $\mathbb{R}$ is Banach space so is $B(X)$ - space of all bounded linear functionals defined on $X$.

Then:

$$\begin{array}{l} \left\| f _ {n} - f _ {n + p} \right\| = \left\| f _ {n} - f _ {n + 1} + f _ {n + 1} - f _ {n + 2} + f _ {n + 2} - \dots + f _ {n + p - 1} - f _ {n + p} \right\| \leq c _ {n} + c _ {n + 1} + \dots + c _ {n + p - 1} \leq \\ \leq \sum_ {k = n} ^ {\infty} c _ {k} \xrightarrow [ n \to \infty ]{} 0 \end{array}$$

So sequence $\left(f_n\right)_{n\in \mathbb{N}}$ is Cauchy sequence, and as $B(X)$ is complete then in converges to some $f\in B(X)$, and so it is uniformly convergent by definition.