Show that any representation of $\mathfrak{sl}(2,\mathbb C)$ is a subrepresentation of $V^{\otimes m} \oplus V^{\otimes {(m+1)}}$ for some $m$ 
Suppose $M$ be a finite-dimensional representation of $\mathfrak{sl}(2,\mathbb C)$, then there is a positive integer $m$ such that $M$ is isomorphic to a subrepresentation of $V^{\otimes m} \oplus V^{\otimes {(m+1)}}$,where $V$ is the standard $2$-dimensional representation of $\mathfrak{sl}(2,\mathbb C)$.

Since $M$ is a finite-dimensional representation of $\mathfrak{sl}(2,\mathbb C)$, by the complete reducibility theorem $ M=\bigoplus\limits_{i=1}^n V(n_i)^{\oplus a_i} \oplus\bigoplus\limits_{j=1}^m V(m_j)^{\oplus b_j}$ where all $a_i$ and $b_j$ are distinct and $a_i$ and $b_j$ are multiplicities of $V(n_i)$ and $V(m_j)$, respectively, and $n_i$'s are odd and $m_j$'s are even.  Any ideas to proceed further? 
 A: Let $L(w)$ denote the simple $\mathfrak{sl}_2(\mathbb{C})$-module with $\mathfrak{h}$-highest weight $w\in\mathbb{C}$ with respect to the Borel subalgebra $\mathfrak{b}$ consisting of upper-triangular matrices, where the Cartan subalgebra $\mathfrak{h}$ is the subalgebra composed by diagonal matrices.  Then, $V=L(1)$.  Observe that $$V^{\otimes n}=\bigoplus_{j=0}^{\left\lfloor\frac{n}{2}\right\rfloor}\,\big(L(n-2j)\big)^{\oplus \left(\binom{n}{j}-\binom{n}{j-1}\right)}$$
for all positive integers $n$.  As you have stated, $M$ can be decomposed as $$M=\left(\bigoplus_{\mu=0}^r\,\big(L\left(a_\mu\right)\big)^{\oplus k_\mu}\right)\oplus\left(\bigoplus_{\nu=0}^s\,\big(L\left(b_\nu\right)\big)^{\oplus l_\nu}\right)$$
where $r,s\in\mathbb{N}_0$, $k_1,k_2,\ldots,k_r\in\mathbb{N}$, $l_1,l_2\ldots,l_s\in\mathbb{N}$, $a_1<a_2<\ldots<a_r$ are nonnegative even integers, and $b_1<b_2<\ldots<b_s$ are nonnegative odd integers.  Now, we can pick $m\in\mathbb{N}$ such that $$m\geq 2+ \max\left\{a_r,b_s,k_1-1,k_2-1,\ldots,k_r-1,l_1-1,l_2-1,\ldots,l_s-1
\right\}\,.$$  Ergo, it follows that $M$ is a direct summand of $V^{\otimes m}\oplus V^{\otimes (m+1)}$.
