How to prove that if ${_}$ is bounded in $(C^0[0,1],‖⋅‖_∞)$, then ${_}$ is bounded in $(^0[0,1],‖⋅‖_∞)$?

Question

Given the following differential equation $u''(t)={f}_{k}(x)$ with initial conditions $u(0)=u'(0)=0$; where ${f}_{k}(x)$∈($C^{0}[0,1],\| \cdot \|_{∞}$).
How can I prove that if {${f}_{k}$} is a bounded sequence in ($C^{0}[0,1],\| \cdot \|_{∞}$), then {${u}_{k}$} is bounded in $(C^{0}[0,1],\| \cdot \|_{∞})$?
I only know that ($C^{0}[0,1],\| \cdot \|_{∞}$) is a complete metric space.. Can someone help me please?

Answer 1

By the fundamental theorem of calculus we have that $$ u_k(t) = \int^t_0 \int_0^s f_k(x)~\mathrm{d}x~ \mathrm{d}s. $$ Hence (note that $0 \leq s \leq 1$): $$ \lVert u_k \rVert_{\infty} \leq \sup_{t \in [0, 1]}\int^t_0 \int^s_0 \lVert f_k \rVert_{\infty} ~\mathrm{d}x ~\mathrm{d}s \leq \lVert f_k \rVert_{\infty} \leq \lVert f_k \rVert_{\infty}$$ This means that $\lVert u_k \rVert_{\infty}$ has at least the same bound as $\lVert f_k \rVert_{\infty}$.