[proofplan]
The proof is a fixed-pair limiting argument. For each pair of distinct points $x,y \in X$, the Hölder estimate for every $f_j$ gives a uniform inequality with constant $M$. [Uniform convergence](/page/Uniform%20Convergence) implies pointwise convergence at both $x$ and $y$, so the Euclidean distance $|f_j(x)-f_j(y)|$ converges to $|f(x)-f(y)|$. Taking the supremum over all distinct pairs gives the Hölder seminorm bound, and the same argument along a seminorm-convergent subsequence gives the sharper liminf estimate.
[/proofplan]
[step:Pass the uniform Hölder estimate to the pointwise limit]
If $X$ has no two distinct points, then by the convention in the statement,
\begin{align*}
[f]_{C^{0,\gamma}(X)} = 0
\end{align*}
and both asserted seminorm estimates follow. Assume therefore that $x,y \in X$ are distinct. Since $f_j \in C^{0,\gamma}(X;\mathbb{R}^m)$, the definition of the Hölder seminorm gives
\begin{align*}
|f_j(x)-f_j(y)| \le [f_j]_{C^{0,\gamma}(X)} |x-y|^\gamma
\end{align*}
for every $j \in \mathbb{N}$. By the definition of $M$,
\begin{align*}
|f_j(x)-f_j(y)| \le M |x-y|^\gamma
\end{align*}
for every $j \in \mathbb{N}$.
Uniform convergence on $X$ implies pointwise convergence at $x$ and at $y$, so
\begin{align*}
f_j(x) \to f(x)
\end{align*}
and
\begin{align*}
f_j(y) \to f(y)
\end{align*}
in $\mathbb{R}^m$. Addition, subtraction, and the Euclidean norm are continuous on finite-dimensional Euclidean spaces, hence
\begin{align*}
|f_j(x)-f_j(y)| \to |f(x)-f(y)|.
\end{align*}
Passing to the limit in the real inequality above gives
\begin{align*}
|f(x)-f(y)| \le M |x-y|^\gamma.
\end{align*}
[guided]
Fix two distinct points $x,y \in X$. The goal is to prove the desired Hölder estimate for this one pair and only then take the supremum over all pairs. For each $j \in \mathbb{N}$, the map $f_j$ belongs to $C^{0,\gamma}(X;\mathbb{R}^m)$, so its Hölder seminorm controls the difference quotient between $x$ and $y$:
\begin{align*}
|f_j(x)-f_j(y)| \le [f_j]_{C^{0,\gamma}(X)} |x-y|^\gamma.
\end{align*}
The hypothesis defining $M$ says that every seminorm $[f_j]_{C^{0,\gamma}(X)}$ is bounded above by $M$. Substituting this bound into the preceding inequality gives
\begin{align*}
|f_j(x)-f_j(y)| \le M |x-y|^\gamma
\end{align*}
for every $j \in \mathbb{N}$.
Now we pass this fixed-pair inequality to the limit. Uniform convergence of $f_j$ to $f$ on $X$ means that
\begin{align*}
\sup_{z \in X} |f_j(z)-f(z)| \to 0.
\end{align*}
In particular, evaluating at the two fixed points $x$ and $y$ gives
\begin{align*}
f_j(x) \to f(x)
\end{align*}
and
\begin{align*}
f_j(y) \to f(y).
\end{align*}
Because subtraction and the Euclidean norm are continuous maps on $\mathbb{R}^m$, this implies
\begin{align*}
|f_j(x)-f_j(y)| \to |f(x)-f(y)|.
\end{align*}
The right-hand side $M|x-y|^\gamma$ does not depend on $j$, so the elementary order property of limits in $\mathbb{R}$ gives
\begin{align*}
|f(x)-f(y)| \le M |x-y|^\gamma.
\end{align*}
This proves the Hölder estimate for the chosen pair $x,y$.
[/guided]
[/step]
[step:Take the supremum over distinct pairs to obtain the Hölder seminorm bound]
For every distinct pair $x,y \in X$, the previous step gives
\begin{align*}
\frac{|f(x)-f(y)|}{|x-y|^\gamma} \le M.
\end{align*}
Taking the supremum over all such pairs yields
\begin{align*}
[f]_{C^{0,\gamma}(X)} \le M.
\end{align*}
Thus $f$ is Hölder continuous with exponent $\gamma$. If the convention for $C^{0,\gamma}(X;\mathbb{R}^m)$ includes boundedness as part of the space, choose $j_0 \in \mathbb{N}$ such that
\begin{align*}
\sup_{x \in X} |f(x)-f_{j_0}(x)| \le 1.
\end{align*}
Since $f_{j_0} \in C^{0,\gamma}(X;\mathbb{R}^m)$ is bounded under that convention, the estimate
\begin{align*}
|f(x)| \le |f(x)-f_{j_0}(x)| + |f_{j_0}(x)|
\end{align*}
shows that $f$ is bounded on $X$. Hence $f \in C^{0,\gamma}(X;\mathbb{R}^m)$ under either convention.
[/step]
[step:Repeat the fixed-pair argument along the liminf-realizing subsequence]
Let $(f_{j_l})_{l \in \mathbb{N}}$ be a subsequence such that
\begin{align*}
[f_{j_l}]_{C^{0,\gamma}(X)} \to L,
\end{align*}
where
\begin{align*}
L := \liminf_{j \to \infty} [f_j]_{C^{0,\gamma}(X)}.
\end{align*}
For distinct $x,y \in X$, the Hölder estimate for $f_{j_l}$ gives
\begin{align*}
|f_{j_l}(x)-f_{j_l}(y)| \le [f_{j_l}]_{C^{0,\gamma}(X)} |x-y|^\gamma
\end{align*}
for every $l \in \mathbb{N}$. Since $f_{j_l} \to f$ uniformly on $X$, the left-hand side converges to $|f(x)-f(y)|$. Since the seminorms converge to $L$, the right-hand side converges to $L|x-y|^\gamma$. Passing to the limit gives
\begin{align*}
|f(x)-f(y)| \le L |x-y|^\gamma.
\end{align*}
Dividing by $|x-y|^\gamma$ and taking the supremum over all distinct $x,y \in X$ yields
\begin{align*}
[f]_{C^{0,\gamma}(X)} \le L.
\end{align*}
Substituting the definition of $L$ gives
\begin{align*}
[f]_{C^{0,\gamma}(X)} \le \liminf_{j \to \infty} [f_j]_{C^{0,\gamma}(X)}.
\end{align*}
This proves the asserted liminf estimate.
[/step]
