Eckart–Young–Mirsky theorem: rank $\le$ k or rank = k

Question

The Eckart–Young–Mirsky theorem is stated sometimes with rank $\le$ k and sometimes with rank = k. Why?

More specifically, why the following two optimization problems are equivalent:

Given a matrix $A \in \mathbb{R}^{n \times d}$, and a natural number $k < $ rank(A).

1. $\min_{B \in \mathbb{R}^{n \times d}, rank(B) \color{red}\le k}||A-B||^2_F$

2. $\min_{B \in \mathbb{R}^{n \times d}, rank(B) \color{red}= k}||A-B||^2_F$