Let $A$ be a local ring and $f: M\rightarrow N$ be a map of finitely generated $A$-modules. Let $f_{\mathfrak{m}}$ be the induced map from $M\otimes A/\mathfrak{m}\rightarrow N\otimes A/\mathfrak{m}$. Assume that $f_{\mathfrak{m}}$ is surjective. I want to show that $f$ is also surjective.
Attempt: Let $x_{1},...,x_{k}$ be generators of $M$ and $y_{1},...,y_{l}$ be generators of $N$. Now let $n\in N$, then $n = \sum_{i=1}^{l}a_{i}y_{i}$ for some $a_{i}\in A$. We can consider the element $(\sum_{i=1}^{l}a_{i}y_{i}\otimes 1)\in N\otimes A/\mathfrak{m}$. By surjectivity of $f_{\mathfrak{m}}$ we have that there exists $\sum_{j=1}^{k}b_{j}x_{j}\otimes \overline{a}\in M\otimes A/\mathfrak{m}$ such that $f_{\mathfrak{m}}(\sum_{j=1}^{k}b_{j}x_{j}\otimes \overline{a}) = (\sum_{i=1}^{l}a_{i}y_{i}\otimes 1)$. Then we must have $f(\sum_{j=1}^{k}b_{j}x_{j})\otimes \overline{a} = (\sum_{i=1}^{l}a_{i}y_{i}\otimes 1)$. From here I am stuck.