We know that there are two definitions to describe lasso.

Regression with constraint definition: $$\min\limits_{\beta} \|y-X\beta\|^2, \sum\limits_{p}|\beta_p|\leq t, \exists t $$ Regression with penalty definition: $$\min\limits_{\beta} \|y-X\beta\|^2+\lambda\sum\limits_{p}|\beta_p|, \exists\lambda$$

But how to convince these two definition are equivalent for some $t$ and $\lambda$? I think Lagrange multipliers is the key to show the relationship between two definitions. However, I failed to work out it rigorously because I assume the properties of lasso ($\sum\limits_{p}|\beta_p|=t$) in regression with constraint definition.

Does anyone can show me the complete and rigorous proof of these two definitions are equivalent for some $t$ and $\lambda$?

Thank you very much if you can help.

EDIT: According to the the comments below, I edited my question.