diff --git a/CHANGELOG.md b/CHANGELOG.md
index dc8e8b799e..c13ba8b8d3 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -304,6 +304,9 @@ The format is based on [Keep a Changelog](http://keepachangelog.com/en/1.0.0/).
 - Fixed process rank not being available right away after `Trainer` instantiation ([#6941](https://github.com/PyTorchLightning/pytorch-lightning/pull/6941))
 
 
+- Fixed `lr_find` trying beyond `num_training` steps and suggesting a too high learning rate ([#7076](https://github.com/PyTorchLightning/pytorch-lightning/pull/7076))
+
+
 - Fixed logger creating incorrect version folder in DDP with repeated `Trainer.fit` calls ([#7077](https://github.com/PyTorchLightning/pytorch-lightning/pull/7077))
 
 
diff --git a/pytorch_lightning/trainer/training_loop.py b/pytorch_lightning/trainer/training_loop.py
index b7f4c58444..79ed39676a 100644
--- a/pytorch_lightning/trainer/training_loop.py
+++ b/pytorch_lightning/trainer/training_loop.py
@@ -521,7 +521,7 @@ class TrainLoop:
 
             # max steps reached, end training
             if (
-                self.trainer.max_steps is not None and self.trainer.max_steps == self.trainer.global_step + 1
+                self.trainer.max_steps is not None and self.trainer.max_steps <= self.trainer.global_step + 1
                 and self._accumulated_batches_reached()
             ):
                 break
diff --git a/pytorch_lightning/tuner/lr_finder.py b/pytorch_lightning/tuner/lr_finder.py
index e3ccef9aa7..2d122a3d30 100644
--- a/pytorch_lightning/tuner/lr_finder.py
+++ b/pytorch_lightning/tuner/lr_finder.py
@@ -180,7 +180,7 @@ def lr_find(
 
     # Prompt if we stopped early
     if trainer.global_step != num_training:
-        log.info('LR finder stopped early due to diverging loss.')
+        log.info(f'LR finder stopped early after {trainer.global_step} steps due to diverging loss.')
 
     # Transfer results from callback to lr finder object
     lr_finder.results.update({'lr': trainer.callbacks[0].lrs, 'loss': trainer.callbacks[0].losses})
diff --git a/tests/trainer/test_lr_finder.py b/tests/tuner/test_lr_finder.py
similarity index 88%
rename from tests/trainer/test_lr_finder.py
rename to tests/tuner/test_lr_finder.py
index 44510eb161..dcefedb4a2 100644
--- a/tests/trainer/test_lr_finder.py
+++ b/tests/tuner/test_lr_finder.py
@@ -295,3 +295,45 @@ def test_lr_find_with_bs_scale(tmpdir):
 
     assert lr != before_lr
     assert isinstance(bs, int)
+
+
+def test_lr_candidates_between_min_and_max(tmpdir):
+    """Test that learning rate candidates are between min_lr and max_lr."""
+    class TestModel(BoringModel):
+        def __init__(self, learning_rate=0.1):
+            super().__init__()
+            self.save_hyperparameters()
+
+    model = TestModel()
+    trainer = Trainer(default_root_dir=tmpdir)
+
+    lr_min = 1e-8
+    lr_max = 1.0
+    lr_finder = trainer.tuner.lr_find(
+        model,
+        max_lr=lr_min,
+        min_lr=lr_max,
+        num_training=3,
+    )
+    lr_candidates = lr_finder.results["lr"]
+    assert all([lr_min <= lr <= lr_max for lr in lr_candidates])
+
+
+def test_lr_finder_ends_before_num_training(tmpdir):
+    """Tests learning rate finder ends before `num_training` steps."""
+    class TestModel(BoringModel):
+        def __init__(self, learning_rate=0.1):
+            super().__init__()
+            self.save_hyperparameters()
+
+        def training_step_end(self, outputs):
+            assert self.global_step < num_training
+            return outputs
+
+    model = TestModel()
+    trainer = Trainer(default_root_dir=tmpdir)
+    num_training = 3
+    _ = trainer.tuner.lr_find(
+        model=model,
+        num_training=num_training,
+    )